What happens when relativistic effects stop?

Calculating the effect of acceleration in special relativity is straightforward, but I suspect the algebra is a bit much at high school level. See John Baez's article on the Relativistic Rocket for a summary, or see Chapter 6 of Gravitation by Misner, Thorne and Wheeler for a more detailed analysis.

When you're first introduced to SR you tend to be told about time dilation and length contraction and given formulae to calculate them. However this is at best an oversimplification and at worst actively misleading. When you're looking at some object moving relative to you you do indeed measure the object's length to be contracted, but what actually happens is that the two end points in the object's rest frame transform into points at slightly different times in your rest frame. You measure the object to be contracted because you're measuring the end points at slightly different times. There is no sense in which the object is squeezed by it's high velocity.

Any object has a proper length, which is equal to its length in its rest frame. The proper length is an invariant and all observers will measure the same proper length regardless of their relative velocity. If you consider proper length then the object is not contracted.

Anyhow, the answer to your question is that when the object comes to a stop relative to you its length has not changed. This is because it never did change - the change you measured was due to the coordinates you were using not matching the coordinates the object was using. When the object comes to rest in your frame you and the object are using the same coordinates (at worst differing in the position of the origin) so both of you measure the length to be the proper length.

Everything remains relativistic. However, the variation between the Newtonian and Einstein relativity is of the order of $v^2/c^2$. So when this number drops below observable error, you can't tell the difference.

If you are the observer and you are observing a car move in front of you, then the clock in the car will appear to move slower than the clock in your reference frame and the length of the car will appear to be contracted to you in the direction of motion. All these effects get magnified as the velocity of the car approaches that of light. In the same way, all these effects get less and less magnified if the velocity decreases and completely stops when there is no relative velocity between you and the car.