1- What is it about Edward and Jake that makes their answer to Bella’s request that they stay behind and not fight different? Would Jake’s answer be different if Edward were not in the picture?
How much their families raised them makes the difference. Jacob's pack had basically taught their members to fight to matter what. If they are in love with a woman who isn't a werewolf, she is expected to wait until they come back... like Emily and Sam. Therefore Jacob felt obligated to go fight, so it wouldn't matter if Edward was in the picture or not. Fighting is almost like a second nature to everyone in the pack. Edward's family, on the other hand, are a lot more understanding with this issue. His family would not comdemn him either way. In fact, they believe that you should fight only when it is absolutely nessary. Therefore, to Edward he had more choice in the sense that if he didn't join in on the fight, he family would not view him as a traitor.
2- Is there something more to why Jake doesn’t step into the role of Alpha Male? Will he decide to become chief later on?
Yes, I think that he was counting on Bella choosing him over Edward and he wanted to have the opportunity to spent more time with her than his pack. If he had nothing to loose, I think he would become chief at some point.
3- Why is the wedding so important to Alice? Is her motivation truly for Bella’s (eventual) happiness or is she being selfish in wanting to coordinate a big event?
Alice knows that she missed out on a lot of human events in her life, so anything a big occasion (or just a fun event) rolls around, she can't help but want to have as much of it as she can. I think it's also because Alice is the type of person who likes to have fun. It gives her an opportunity to take a break from the constant chaos in the Cullen family. (By chaos I mean there is always some war that they have to prepare for, or some future to watch. In a way, Alice is the one who is under the most pressure because she is expected to be perfect with her visions). As for Bella's happiness, honestly no I don't think that is really her motivation. Alice can add spunk to the event, but Bella is a hard person to make happy. She always has something that occupies her mind.