Can I use these interchangeably?
That doesn't give you immunity.
That does not grant you immunity.

That doesn't excuse what you did.
That doesn't justify what you did.

And this sentence OK?
It doesn't warrant that kind of behaviour.

Question

Can I use these interchangeably?
That doesn't give you immunity. 
That does not grant you immunity.

That doesn't excuse what you did.
That doesn't justify what you did.

And this sentence OK?
It doesn't warrant that kind of behaviour.

Answer

Yes.
excuse and justify are a little different, but you can interchange them here without changing anything significant.

Answer

"That doesn't give you immunity."
"That does not grant you immunity."
Yes, I believe these two can be used interchangeably.

"That doesn't excuse what you did." - To excuse means to give a reason to lessen a negative situation. This sentence is saying 'That does not make what you did any better, what you did was still bad.'
"That doesn't justify what you did." - To justify is like giving a logical/moral reason to a negative situation. So this sentence is saying 'that does not mean what you did was the right thing to do. You are still wrong for doing it.'
You can use them interchangeably, but they won't really mean the same thing.