C#: should object variables be assigned to null?

What matters more IMO is to call Dispose on objects that implement IDisposable.

Apart from that, assigning null to reference variables will just mean that you are explicitly indicating the end of scope — most of the time, it's just a few instructions early (for example, local variables in the method body) — with the era of compiler/JIT optimizations, its quite possible that runtime would do the same, so you really don't get anything out of it. In a few cases, such as static variables etc, whose scope is application level, you should assign a variable to null if you are done using it so that object will get garbage collected.

Should you turn off your car before pushing it to the lake?
No. It is a common mistake, but it doesn't make any difference. You aren't setting the object to null, just one reference to it - the object is still in memory, and must still be collected by the garbage collector.

No, and that could in fact be dangerous and bug-prone (consider the possibility that someone might try to use it later on, not realizing it had been set to null). Only set something to null if there's a logical reason to set it to null.