Add extra stud to the wall etc.

[timmerhuset_78]

Hello everyone!
I'm in need of some answers and advice.
As a first-time homeowner, you want a bit of reassurance before making too many changes.

I had a broken plaster wall in the closet (towards the outer wall) that I was going to replace with OSB and plasterboard.
Turns out there was no stud at the far left of the room. The old plasterboard continues behind the next wall.

And now for the questions!

1. Would it be easiest to put up an extra stud on the far left?
2. If I add an extra stud, can I cut the brown felt and just push the insulation inward with the stud?
3. This brown "paper felt," what's its name? I have the same under the insulation in the attic.
4. It has some damage here and there. Should I attempt to patch it in some way?

Let's see if I made myself understood.

 

[KnockOnWood]

Interior wall or exterior wall?
If it's a plastic film that you've cut a hole in and it’s an exterior wall, you should probably "fix" it.
There is age-resistant plastic tape available "everywhere."

 

[timmerhuset_78]

Interior wall or exterior wall?
If it's a plastic film that you've cut a hole in and it's an exterior wall, you should probably "repair" it.
You can buy aging-resistant plastic tape "everywhere".

It's an exterior wall but it's not plastic. More like "felt" or paper.
The previous owners had driven long screws far into the insulation. I assume it's an old vapor barrier but don't really know what this is called.

 

[timmerhuset_78]

It's the outer wall but it's not plastic. More like "felt" or paper.
The previous owners had driven long screws deep into the insulation. I assume it's some kind of old vapor barrier but I don't really know what it's called.

it's also not completely sealed, just overlapping