The Core Set Lannister deck, as is, is not particularly well suited to the multiplayer game. They are, however, very good in one-on-one, particularly once you start using multiples of the key cards instead of relying on the luck of the draw. The other thing, particularly for people who are new to the game, is that Lannister is the House with the most "moving parts." Individual cards may not seem as strong, but they have a lot of cards that work together and end up being more than the "sum of the parts." They're hard to play, though, because you have to REMEMBER all of those moving parts to get the full effect of their sneakiness/strategy.
In our group, Lanni pretty well rules the LCG-Joust. Not that it always wins, but it always lets the other player know they were in a fight.
On your #4 question, here is the basic anatomy of a challenge:
1) Before the active player attacks, all players take turns (going clockwise from the First Player) triggering player actions (Any Phase or Challenges effects) one at a time. When no one has any more to use, the active player may attack.
2) The active player attacks by a) announcing the challenge type and opponent, and b) declaring/kneeling their attackers. Nothing can be done between a) and b), including passive effects.
3) As in #1, players take turns triggering Any Phase or Challenges effects one at a time until no one has any more they want to do.
4) Defense of the challenge is determined by a) the attacking player choosing Stealth targets (if an attacking character has Stealth), and b) the defending player declaring/kneeling their defenders (unless the choose to let the challenge go unopposed). Again, nothing can be done between a) and b), including passive effects.
5) As in #1 and #3, players take turns triggering Any Phase or Challenges effects one at a time until no one has any more they want to do.
6) The challenge is resolved by a) determining who wins by adding and comparing the total attacking STR to the total defending STR, b) settling claim, c) awarding power if the challenge was unopposed and d) awarding power for Renown if any characters on the winning side have it. Nothing can be done between a), b), c) or d) (except for "save" Responses; "save" and "cancel" Responses are the only things in the game that can interrupt things), including passive effects. Passives and Responses to anything that happens in a-d is held until after d) is complete.
7) Go back to #1 for the next challenge.
So the answer to your question ("For any phase abilities, could you use them after defenders are declared?") the short answer is "yes," but there is more to the timing rules than that. The above "anatomy" may help keep things straight.