Are both sentences OK?
I felt something inside me break after he left.
I felt something inside me tear away after he left.And these?
Let's have some rest.
Let's take a short rest.

Question

Are both sentences OK?
I felt something inside me break after he left.
I felt something inside me tear away after he left.And these?
Let's have some rest.
Let's take a short rest.

Answer

You can also say, "Let's rest" or "Let's rest for a few minutes". 
Let's take a short rest is correct or let's take a short break sound even more natural.

Answer

They are all correct, but native English speakers are more likely to say, "Let's take a short rest" than "Let's have some rest."