What will the last days of Earth be like? The sun is a main-sequence star; as I understand it the sun will expand into a red giant in a few billion years.  I have heard conflicting reports of what this entails - some sources I found say the sun will expand until the current orbit of Earth is contained within it, others say it will stop expanding short of Earth's current orbit.

Every source has agreed that the planet will have long-since stopped being able to support life as we know it.

Assuming humanity takes no action to significantly alter the natural course of events, such as dramatic alterations of our atmosphere, what will the last few decades of Earth's life be like?  Will the planet be heated to the point where its surface is like Mercury's sun-side?  Will the sun's output drop low enough that it instead cools, like Mars?  Will the sun expand and engulf the planet, or will the planet continue to orbit an expanded sun as a lifeless ball?

Will the sun's expansion be gradual or rapid?  How will this expansion affect the rocky planets nearest to it?
A lot of the answer depends on Orbital mechanics and calculating that isn't easy.   If no action is taken, Earth's orbit is still likely to change, but how much and in which direction is hard to say.

Long before the sun goes red giant it will get too hot for the Earth.  If the earth doesn't move, the sun's increased temperature will likely make the earth uncomfortable in a few hundred million years and perhaps scorching in as little as a billion years.   Warmer air and more water vapor in the air will trap heat and at a certain point, you'd get a run-away green house effect, long before the red-giant phase.    Short term, changes in CO2 in the atmosphere might slow that.   We could see ice ages before we see run-away heating, but over time, with no human action, run-away heating is likely inevitable - on a very large timescale, like a billion years or so.   Longer if the Earth drifts farther from the sun.


  what will the last few decades of Earth's life be like? Will the
  planet be heated to the point where its surface is like Mercury's
  sun-side? Will the sun's output drop low enough that it instead cools,
  like Mars? Will the sun expand and engulf the planet, or will the
  planet continue to orbit an expanded sun as a lifeless ball?
  
  Will the sun's expansion be gradual or rapid? How will this expansion
  affect the rocky planets nearest to it?


I'm going to answer the 2nd part first.   The sun's expansion will first be gradual, but the red giant stage will be fairly rapid.   Over the next 4 billion years or so, the sun will get some 60% more luminous, so significantly much more heat at the same orbit.   A very rough rule of thumb is 10% more luminous every billion years. (that's not for every star, just for 1 solar mass) - and that's a pretty ballpark estimate.

But just 10% more heat from the sun would raise the Earth's temperature uncomfortably, and with the added water vapor in the air and greenhouse effect, the earth might be a permanent steam room, even at the poles, at just 10% more heat from the sun.

asking about the last few decades is the wrong time frame as changes would probably happen much more slowly, though atmospheric/feedback mechanisms in the climate might make changes happen more quickly - ice age speed for example, a few thousand years or so.   Also, as Antarctica drifts away from the south pole, we might see an end to ice ages - there's lots of moving parts to long term climate predictions, but in general, you'd see gradual heating to the point where complex life might have a hard time adapting.  

The suns output won't drop until after it goes red giant, it's on a steady but gradual increase, followed by a relatively fast increase at the red giant stage.

Now, if we terraform Titan, that might be a safe distance to watch the sun go red giant from and it would probably be quite a spectacular show, well, over enough time.

As to how fast it will go red-giant, fast for a star means, about 200 million years, so, from the perspective of a human lifetime, watching a star go red giant would be like watching grass grow.

A much more intelligent discussion on that than I can give, here:   https://physics.stackexchange.com/questions/25622/how-fast-will-the-sun-become-a-red-giant