How accurate are rederings of something entering Earth's atmosphere? Supposedly how accurate is it when you see a movies showing something entering Earth's atmosphere traveling 1 mile to 10 miles an hour or barely at all but skinning the atmosphere they show it heating up and burning just by entering the atmosphere no speed or barely just entering?

As a second question, something coming in - say, a vehicle with rockets - to ensure that the speed does not exceed that would cause  any heating at all. What speed would that be?
In the movies you reference, I doubt that it was ever the intention of a movie maker to suggest that the craft was moving at a crawl when it began to enter the atmosphere. If the craft doesn't look like it is moving very fast, the idea is probably that the point of reference (the "camera") is moving at nearly the same speed as the craft, or perhaps that the craft is so huge, that the reference point is miles away. Everything is relative.

There is essentially no realistic scenario where any object would enter the atmosphere at such a slow speed. Imagine, for example that you have an object, like Felix Baumgartner or something, which is standing approximately stationary in space above the Earth's atmosphere. Gravity will immediately begin pulling on that object and because there is no atmosphere to slow the object down yet, it just keeps accelerating as it falls. Even as it enters the edge of the atmosphere, it doesn't slow down much, because there isn't much atmosphere yet. Felix was moving more than 800 mph before he started to slow down.

The International Space Station is another object that's sort of "stationary", in a manner of speaking. It's not going anywhere because it's in a stable orbit (sort of). But in order to stay up there, it's actually moving more than 17,000 mph, relative to the atmosphere. If something is "stationary" with respect to the sun, just sitting there in Earth's orbit as Earth comes at it, the Earth's atmosphere would smack it at 67,000 mph. That's how fast Earth orbits the sun. So there's really isn't anything that would just gradually drift into the atmosphere at 1 to 10 mph.

If a craft were to use rockets to halt itself before entry, as you suggest, remember Felix again; it can become stationary with respect to the Earth, but then gravity begins its work, so if the craft wants to remain at a very slow speed through the entire decent, it has to keep firing those rockets all the way down. And certainly, nothing would burn up at that pace, other than a lot of rocket fuel. What causes the burning is not something special and hot about the atmosphere, but the fact that the object is smashing into the air (thin as it may be at that altitude) at thousands of mile per hour. It is even theoretically possible for meteors to enter at such speeds that the atoms are literally forced to hit each other's nuclei, and a part of the fire generated would actually a brief nuclear reaction.

Response to Question Edit:
(Thank you HDE. The initial question was very unclear.)
The speed at which friction completely stops is 0. Any movement at all, even in that 1 - 10 mph range, produces a small amount of heat from friction, even if the medium you are moving through is very thin air (though certainly nothing one would characterize as "burning"). You seem to be looking for the speed at which that friction would be low enough to not cause a problem, but there's no magic answer. It depends on the situation. How delecate is the craft? How much wear is acceptable? Essentially, this question becomes something very similar to asking "How fast can I drive with something sticking out the window?" You are trying to keep the entry speed low enough that the wind is not a problem for you as the density of that wind gradually increases from practically nothing to surface air density. If your craft has a heavy, durable heat sheild, properly oriented, then extraordinary speeds are still not a real problem. But if you are trying to create a sand mandala on the roof of your craft as you descend, then you are going to want to control that wind level very carefully.