Somebody in my Legion believes that the Paradox Generator reset his Ability timer. However, it doesn't say it would, so he didn't know to use the ability. Does it reset abilities, or did he just miscalculate?

Actually, he's pretty sure he was just mistaken.

paradox genorators work as time manipulators but also reset the daily reward, as read about in galaxy legion wiki under mission artifacts

potential for free power leveling? (only if you are lucky and keep on getting tri-max or energy cube).

No idea but it is a great question! In theory, it should since it resets all timers.