Is there a way to know how much "influence" you have in Reporting bad questions?

There is no public display of that information. If you have been on the site for a while and only report things that are truly a violation, this increases your reliability. If you report things that are later overturned as violations (successfully appealed) or if you report a single person more than 4-5 times in a row, your reliability will drop. It can easily be regained again, simply by reporting reliably again.

When you report, if things tend to disappear pretty quickly, this is some indication that you have a bit of influence, although it is no guarantee, since others are probably reporting it as well.

Conversely, if you report something and NOTHING happens at all, that might be an indication that your reliability has dropped.

Sure. You need influence on a totally anonymous board.

You can see if the Q or A that you reported is gone. :)