The Three Elements in the Milankovitch Cycles According to the Serbian geophysicist Milutin Milankovitch, there are three elements that make an ice age possible:


Eccentricity (orbital shape):  Varying between 0.000055 and 0.0679 over the course of 100,000 years (1.0 being a perfect circle).
Obliquity (axial tilt):  Varying between 22.1 and 24.5 degrees over the course of 41,000 years.
Axial precession (change in the orientation of the rotational axis on a rotating body):  Polaris being the North Star for a total of 26,000 degrees.


The question is how connected those three elements are, in the event someone wants to change the numbers in a worldbuilding process to make either a longer or shorter ice age:

Does the duration and extent of eccentricity affect those of obliquity and precession?

Or does the duration and extent of obliquity affect those of eccentricity and precession?

Or does the duration and extent of precession affect those of eccentricity and obliquity?
The primary driver of ice ages over the history of the Earth is much more Earth science than astronomy.  The modern ice age cycle (last 2.6 million years or so) is only possible with relatively low levels of CO2 in the atmosphere and land masses near the North Pole where glaciers can form.  Other factors in the modern ice age period are thought to be the formation of the Isthmus of Panama, perhaps the relatively permanent antarctic ice sheet, and just maybe, the formation of the Himalayas as well.   

The set of circumstances required on Earth that make milankovich driven Ice ages possible are pretty complicated and have to be set just about right.  When they are just right, then the Milankovich cycles, which are quite weak, can take over and be the drivers of ice ages.   Prior to 2.6 million years ago, the last time the Earth went through a period of rising and receding ice ages was 440 million years ago when most of the land was on the South Pole.  CO2 was much more abundant, but the single continent sitting over the south pole made the formation of ice sheets possible.   Over most of Earth's history, the Milankovich cycles haven't governed ice ages.

That said, onto the astronomy bits of this question:


  The question is how connected those three elements are, in the event
  someone wants to change the numbers in a worldbuilding process to make
  either a longer or shorter ice age:


There's a strong correlation between the Milankovich cycles and the length of recent ice ages, so the answer here is yes, if you were able to adjust the "numbers" then the length of ice ages would change accordingly.

The 3 Milankovich cycles combine into two primary patterns, the 40,000 year cycle and the 100,000 year cycle (more on that shortly).  These were observed first by geologists and later tied to Milankovich cycles.  What starts and stops ice ages is complicated and there's a few moving parts, so there's not 100% consensus on Milankovich cycles driving ice ages, but the correlation is strong and as far as I've read, this idea is largely agreed upon.

Ice age Cycles

From about 2.3 million years ago to about 800,000 years ago, the 40,000 year ice age cycle was dominant, and for the last 800,000 years, the 100,000 year cycle has been dominant.  The 40,000 year cycle is thought to be the stronger of the two, and one theory as to why the 100,000 year cycle became dominant 800,000 years ago is that the Earth grew slightly cooler, perhaps with the glaciation of Greenland.   See 100,000 year problem.  The 40,000 year cycle still plays a role even within the 100,000 year cycle currently the dominant one.  Article on that here.

Axial tilt.

Axial tilt is on a pretty consistent 41,040 year cycle and it's strongly tied to the arguably somewhat misnamed, 40,000 year cycle.

Axial tilt is the cause of seasons on Earth, so when the Axial tilt is high (over 24 degrees), the seasonal variation increases, meaning, warmer summers and colder winters on both hemispheres.  When it's low (below 23), it means more mild seasons.  While this doesn't change how much sunlight hits the earth, it does change climate on Earth for other reasons,  The overall result is that the more mild seasons lead to a cooler earth.

From Wikipedia  


  This trend in forcing, by itself, tends to make winters warmer and
  summers colder (i.e. milder seasons), as well as cause an overall
  cooling trend.


The reason for this, briefly, is that it still snows in high latitudes in the Northern Hemisphere during mild winters and snow and Ice takes longer to melt during mild summers.  If this persists over enough years, Snow cover can build up, which causes Albedo to rise and reflect more sunlight back into space, and less summer melting of ice, over many years, can lead to an ice age.  Once an ice age gets going it's self perpetuating, sometimes referred to as feedback mechanisms. 

Axial Tilt reached a maximum about 8,700 BCE and it will reach a minimum in about 11,800 CE.  The warming of the Earth from the Axial tilt maximum isn't sudden though, it begins several thousand years before the maximum and continues for several thousand years after.   This fits nicely with the ending of the last ice age around 10-11,000 BCE.  We're currently in a mid-point in the Axial tilt cycle and it's gradually approaching a cooling effect, though, at least for now, the effect of man made climate change is far bigger than the gradual Axial Tilt cooling.

Axial Precession

Precession is a bit more complicated.  The Earth is on a 26,000 year "wobble",  that moves the North Star and other stars around in a circle in the night sky, but what matters with this Milankovich cycle is what month the Earth is closest to the sun, or, at Perihelion.  The change in Perihelion over time works on roughly a 21,000 year cycle, because it combines Axial precession with Apsidal precession.

Currently the Earth is closest to the Sun in January, but that will gradually change and in about 10,500 years, the Earth will be closest to the Sun in July.  When the Sun is closer, the Earth gets slightly warmer, so the current position of this cycle leads to milder seasons in the Northern Hemisphere (which can lead to cooling) and more extreme seasons in the Southern Hemisphere.   Unlike Axial Tilt which affects both hemispheres in the same way, precession affects each hemisphere in the opposite way to the other.   This, in effect either adds to Axial Tilt's effect or subtracts from it.   As pointed out above, the simplest factor is that milder seasons in the Northern Hemisphere that are ice-age friendly and more extreme seasons are not.   So when perihelion is in January, that somewhat cools the Earth and when it's in July it somewhat warms it.  We're currently in a "cold" period of this cycle, we were in a warm one 10,500 years ago, which also times well with the end of the last ice age. This cycle is moving towards a gradual warming and more extreme Northern Hemisphere seasons currently working in opposition to the Axial tilt effect.  It's cycle is roughly twice as fast and so this also ties in nicely with the 40,000 year cycle too, as these 2 periods (21,000 years and 41,000 years) are close to in sync.  These two cycles are thought to be key factors in driving the greening and droughting of the Sahara desert.  Article here and here.

Eccentricity

The Earth's orbital period and semi-major axis of it's orbit remain roughly unchanged.   What changes with eccentricity is how far the focal point (the Sun is at one of the focal points of the ellipse), moves from the center of the elipse.   The effect of the Earth's orbital eccentricity changing is two-fold.  First, Axial precession only matters if the Earth's orbit is an ellipse.   If the Earth's orbit is circular, the precession Milankovich cycle has no effect.  Precession has maximum effect when the Earth's orbit is at it's most elliptical.

The 2nd effect is that a more elliptical orbit means that, because planets move more slowly in their orbit when they are further from the sun (Equal areas in equal time, per Kepler's law), a more elliptical orbit means that over the calendar year, the Earth gets less sunlight because it's average distance form the sun over time is slightly higher.  So, low eccentricity means slightly warmer, high eccentricity, slightly colder, even though the planet passes through an entire orbit every year, the total energy the Earth receives does drop slightly when the Earth's orbit is most eliptical.

In addition, greater eccentricity amplifies the effect of the Axial precession, which when it's at it's highest, tends towards ending ice ages, so this effect operates in both directions, which is part of the reason why the 100,000 year cycle is smaller.

In effect, eccentricity multiplies the effect of precession, but it has an additional effect that works in opposition to the precession ice age driver.  (Does that make sense? - I can try to clean up if not).

Precession and Axial Tilt can add together or cancel each other out.  

and there's other effects.  How much sunlight is reflected off the high albedo south pole, how much sunlight warms the antarctic ocean, how strong the Atlantic current is.   The entire picture is hugely complicated, but from an astronomical point of view, eccentricity is the only one that can reduce or increase total solar energy hitting the Earth.   Axial tilt and precession have no effect on how much sunlight hits the earth, they only drive the variation between the seasons.  


  Does the duration and extent of eccentricity affect those of obliquity
  and precession?


The extent of eccentricity directly increases the seasonal variation effect of precession.  


  Or does the duration and extent of obliquity affect those of
  eccentricity and precession?


No, the obliquity doesn't affect either of those, but the effect of obliquity can add to, or cancel out the effect of precession.


  Or does the duration and extent of precession affect those of
  eccentricity and obliquity?


Precession doesn't effect eccentricity (which way the Earth is pointed obviously doesn't effect eccentricity), but precession can add to or subtract from Obliquity.

That's the gist of it.  

You can run some numbers if you like, but that gets even longer.  The sunlight energy that reaches the earth and how it varies by axial tilt is longitude dependent, so any calculations on changes in energy by changes in axial tilt will vary based on longitude.  At many latitudes, the summer solar energy is more than double what it is for winter.  Even a one or two degree in the angle of axial tilt variation can make a measurable difference in the seasonal variation.

Precession energy variation isn't longitude dependent, but it varies every day with the earth's orbit.   Some bad math on it, with the current eccentricity of about 0.0167, what that means is that at perihelion, the Earth is about .9833 AU from the sun and at aphelion, about 1.0167 AU. Because solar energy diminishes with the square of the distance, the roughly 3.4% variation in distance from the sun works out to about 7% more solar energy hitting the Earth perihelion to anhelion.   When the Elliptical orbit peaks, that ratio exceeds 20% and when it's minimized, less than 2%, but those are ratios by season and location, the total energy the Earth receives from the sun doesn't change, only seasonal variability changes.

That's as far as I'll go with the numbers, as far as the timing, that's pretty straight forward, some of that covered above:

Axial Tilt, when at a maximum it tends to melt glaciers and leads to warming, this last happened 10,700 years ago and fits with the end of the last ice age.   Minimal tilt can lead to cooling and the earth will enter an Axial Tilt Minimum in about 9,800 years.   The entire cycle, high to low back to high lasts 41,040 years.

Precession is significantly amplified with high eccentricity and significantly reduced with low eccentricity.   Earth is currently in a low eccentricity so the precession effect for the last 10,000 years or so has been relatively small.  10,500 years from now precession will be at peak seasonal variation in the Northern Hemisphere, which will work against the Axial Tilt's cooling.   In about 21,000 years, Precession will again be working towards more mild seasons in the Northern Hemisphere, eccentricity will be high and the Axial Tilt will be past it's minimum, but still in a low point.   So, taken together, we might see an ice age begin to form somewhere around 18-20,000 years in the future, but we have little to worry about.  It's not difficult to generate enough atmospheric CO2 to prevent that ice age from happening.   We're already well over the necessary amount of CO2 to prevent a coming ice age.

Finally, Eccentricity was at a minimum about 9,000 years ago and will reach a maximum in about 27,000 years in the future.  Eccentricity is the least consistent of the cycles, it varies between about 90,000 years and 125,000 years and there's a larger, 400,000 year cycle on top of that.

If parts of that aren't clear, I can try to tidy up, or just ask.