The notification status pull down bar is not working because the very upper part of the screen seems damaged (verified running diagnostics that pixels there are not active).
1) Can I see the notifications some other way?
2) Can I assume that screen replacement solves this?
Htc one X phone
If you use a third party launcher you can use gestures for certain functions.
For instance, using Nova Launcher Prime, when I swiped down anywhere on my home screen it would bring down my notifications.
So that could solve that issue for you.
There should be other free alternatives that perform the same function
To answer your last question, a screen replacement should fix the issue though.
Q & A