It is definitely correct English. Problem is, there are different ways to interpret the sentence depending on the stress you put on the words. There’s definitely one way I’d be leaning towards though.

It could mean that you don’t want to do it if this person is present in the room. This is how I would read it.

It could also mean however that you simply don’t want to do whatever it is in this specific room, in which case the ‘with him’ isn’t really necessary information. That is partly why I don’t think this is the correct reading, as well as the fact that the more natural way to get that idea across would be to say:

“I don’t want to do it in here with him.”

So as for Q2, I would also say yes