To me "get rest" in many situations, but not all, sounds more as if it will require a little more mindfulness to rest. I am so overworked these days. I need to rest. (simply I need to relax) I am so overworked these days. I need to get some rest. (I need to find a way to find the time/ conditions to be able to rest.) Please not the insertion of "some" in get some rest. We use this as an imperative sentence as well. Please get some rest. Please try to get some rest. ("Try" here emphasizes the speaker's understanding that it will take a conscious effort to do so.)

I might say "I need some rest".Or I might say "I need to get some rest." It really doesn't matter which way you say it.

I can't think of any situation where it would make a difference which one you used. Both work. However, most people say, "to get some rest", instead of " to get rest ".

You could use either one. I don't believe there is a time to say one over the other. If I was talking to you I would say, " You need to rest." or I could say, "I need to rest." I could also say, "I need to get rest." It would sound more like a complete sentence if I said,"I need to get some rest" The word some acts as a determiner. It speaks of a unspecified amount time.