When should an object to put in a "resting" state for physics?
I have read in the past that for physics, after a certain values (such as velocity) hit a very small range that the object should be put in a "resting" state. How should this be done and when should it be done? Or, is this just a bad technique?
From what I recall from my reading some time ago, there was a potential scenario when it was determined there would be a collision and the collision time was some extraordinarily small value (such as 0.00001f). However, this seems like it could present a false positive, is this true?
It's a rather vague term and it totally depends on the requirements of your physics simulation. That said, it can be a very efficient optimization to let objects 'rest' if you know it won't hurt.
Many physics engines have a so called "freezing" state, which simply means they won't process frozen bodies until certain conditions occur (or not at all). "Resting" may also mean that dynamic bodies are (with velocity approaching zero) treated as static bodies, which are usually faster since no spatial data structures need to be updated.
How should this be done and when should it be done?
When implementing this, you'll almost certainly need to do some experimenting to tweak the parameters, don't expect it to work immediately.
"How" is a question that cannot be answered without further details (i.e. which physics engine, framework, own development ...?).