Does coiling and straightening a wire change its resistance?

I wanted to know first off why the coils are coiled?

Suppose the wire is 10 m long. If you don't coil it, some of the heat it produces is "here" and some of the heat is 10 m away. Coiling it means you can heat a small area instead a long skinny area 10 m long.

when the coils are stretched too far apart thy run cold and when they are close together the run hot. What is the reason for this if resistance doesn’t change whether a wire is could or straight?

The temperature of the coils depends non only on how much heat they produce (\$I^2R\$) but also how much heat they lose to the environment. If you stretch the coil, it has an overall larger surface area over which heat is carried away by conduction and convection. If you compress the coil, it loses heat over a smaller area, and much of the heat produced by one turn of the coil actually heats the neighboring turns, rather than being lost to the environment.

In addition to the accepted answer, coils also offer physical advantages in taking up the change in length when heated without sagging. The wire becomes brittle after use so the spring in the coil makes it easier to reroute into the channel in the firebrick if a coil pops out (heat the wire up when you do this).

I think sharp bends are subject to more strain with heating/cooling cycles so coils avoid those failure spots.

In addition to all of the correct answers: coiling a wire does change its inductance, which is something like resistance except it only affects the flow of AC current, not DC. This isn't the reason for your coils (which are fed from DC or 50/60 Hz AC — at those frequencies, the inductance isn't enough to matter much, and doesn't contribute to heating). However, it is a reason why you will see coils in other kinds of electronics, including radios, motors, and power supplies. They're not trying to keep heat in (usually they want to get rid of as much heat as possible), but they are trying to regulate the flow of current by storing energy in magnetic fields.