I don't know a definitive answer for 1. We play it by checking the game state at the moment the card is resolved in the normal sequence from top to bottom and left to right. So this is before the influence actions. If there is a happening card which allows a change of influence and which is to the left of the card you mentioned, we would resolve that card first (but I don't recall if there is such a card).
There are two of these, one for government and one for enterprise. The precise language on both (substituting Government in the proper spot) is:
"The player(s) with the least Enterprise influence may not gain prestige this turn."
It is an Effect. Effects are intended to be "baked" at the moment the card is resolved in the Action Resolution. These generally fit two categories, one that makes a change to the game state at that moment (Player with most prestige loses 1, etc.) *or* they very clearly indicate a contingent state that when something happens later in the turn, something else happens (Players who publish this turn may ignore influence requirements).
If we ever do a reprint, the language will be changed to make it clear that it is a snapshot and cannot change. This might actually result in the very effect changing, as there is a bit of light tracking to this that I'd rather not have at issue.
For the game as published it should be played as intended: whoever is last (or tied for last) when this card is reached during the Action Resolution phase - which precedes gaining influence - may not gain prestige that turn.