1. Advent are supposed to be the main "culture" faction, what does that exactly mean? Is their culture actually stronger, always pushing the TEC/Vasari one, or its more about having the cultural upgrades/buildings sooner and cheaper? If i play for Vasari, max my culture research and build same number of those media hubs as the Advent Player (with maxed culture himself) on the adjacent planet, will it be a cultural "tie" or will his culture steamroll mine anyway, as its stronger by definition?
Advent culture is NOT actually stronger, its more about having the upgrades sooner (and more of them). One TEC or Vasari culture/media tower has the same strength as one Advent culture tower (ie: '20 Culture'). But opposing towers won't be a "tie" either, because culture repels much easier than it overtakes. Culture leaving a planet is divided by the number of phase lanes leaving, whereas culture entering a planet has to overcome the full strength of the tower there. In other words, culture is very easy to counter. 1 culture tower will usually counter 3 or more adjoining hostile towers!
For example: if you have 4 planets in a straight line : A--B--C--D (and unupgraded culture), it will be a "tie" between 2 towers on B, against 1 tower on C; but visa versa the one tower on C is only sending out '10' against the '40' defending B. 1 tower on each would also be a stalemate, since they are both sending out "10", but defending with the full "20". It would take 3 towers on B (=30), to overwhelm the 1 tower on C (=20).
If the planets have lot of branching (phase lanes) rather than a straight line, it requires even more towers to overcome 1 defending tower. (Note: Now you can 'tech up' culture output. And Advent Starbases can put out "35" culture, with each upgrade).
2. Should i build the communion temples at my border worlds primarily? I mean, because the culture rates get weaker with the number of jumps, its best to build them closest to the enemy, right?
At the start of the game, No! But later, maybe yes. Only one tower at your homeworld can eventually raise the allegiance of your entire empire. Later, you might need to repel culture, secure uncolonized planets, or attempt a hostile take over from your border worlds.