Welcome! Let us know what questions you have. We're always happy to help.
Be warned, though, that some of us have been playing for a long time, and when you ask a question, we might also answer a bunch of extra questions you didn't ask or bring up card-specific exceptions that you haven't seen yet. So if there are times when we come across as a college calculus professor making algebra waaaay too complicated because we know what's ahead of you, please forgive.
1. When using Robb Stark and you win the military challenge do you choose and kill one of the losing players characters even if that character did not participate in the challenge? Just to be sure, you choose after your opponent chooses who dies from the challenge right?
Correct. Since Robb says to choose a character, without limiting that to a "participating character" or a "kneeling character" or something like that, you can choose absolutely any character controlled by that losing player.
You are also correct about claim effects coming first. As a general rule, settling claim (killing a character for military, discarding a card for intrigue, or stealing power for power) will ALWAYS happen before any "response" effect to winning/losing/resolving the challenge can be used.
2. When my opponent has Viserys or Ser Davos in play can they always choose him if I win a military challenge? This seems over powered because he always returns to their hand and is played again next turn.
That is true. They can choose him to die for military claim, take them back into hand, play him again next round, and do it all over again. It is powerful, but it isn't as over-powered as it seems at first glance. For one, they have to keep PAYING for them, which means they are not paying for other things that get them closer to winning themselves. But yes, when playing with just the Core Set, that can get pretty annoying to play against.
3. When using a card like Strong Belwas can I choose to save a character that was chosen as the killed character because of a military challenge?
Yes. When a character says "save from being killed," they can save the character from any kill effect, even military claim. (The exception, of course, is if the kill effect specifically says "cannot be saved".)
4. If I have a location like Great Hall can I kneel it to make a Baratheon character cost 0?
Yes. There is no "minimum cost" for characters, locations, etc. If you have enough reducers to lower the cost to "free," awesome!
5. When playing Lion's Will can I pay 4 gold to kneel two characters?
No. When you trigger an effect, you can only do what it written. You cannot assume variables or flexibility where there is none. So "pay 2 gold to choose and kneel a character" doesn't give you any room to increase or decrease the number of gold or the number of characters knelt. If you had two copies of the event in your hand, you could play both copies, one after the other, so that it worked out to "pay 4 to kneel 2," but you cannot arbitrarily multiply the single event card by 2.
6. If I have Lightbringer attached to a character can I kneel it instead of killing the attached character for a lost military challenge. This seems over powered as well.
Yes. This is just like the question about Strong Belwas. It is strong, but again, not as overpowered as you might think at first glance. For one thing, people will pile up the kills on that one character until it is gone. For another, think some of the "discard an attachment" effects that are in the Targaryen deck. True, when playing in with just the Core set, it can be annoying, though.