Does the upper limit switch act as a breaker? Will the red button pop up only when it needs to be reset? In other words, if the red button is not popped up does that mean it is working correctly?

If the red button does not pop up it usually means that the furnace has not overheated and caused it to pop. They do get old and pop before the prescribed temp limit and sometimes they fail shut. Under no circumstances should this switch be bypassd. If in doupt replace the part with the correct one. Normally this red button should never have to be pushed in under normal circumstances. If it has popped there must be a reason.