I will not try PVP because it's based on scheduling and I can't be there just anytime they schedule a battle; my sister played and she had an appointed time she had to be there . . . am 'on call', the phone rings, I have to go and I'm just a grandma . . .
So, unless I'm misunderstanding how it works and haven't found anything telling me any different, I can't even try it let alone join a team.
No, tournaments grant you arena tickets, and other loot bonuses when you finish them. It's a quick way to get a lot of tickets if you win. However, if you are looking to get a lot of rank, you need to do ranked pvp to do it. You can also try practice to do deck setups, gear setups, and other strategies.
Tournaments are a great way to get quick arena tickets, and other cool rewards. With that being said, the only way to get rank is through Ranked pvp. However, if you're just looking for fast tickets, tournaments are the way to do it. You can also try Practice pvp, if you're trying to learn new setups, for your deck, gear, or other strategies. Practice pvp does not take/give any rank, and doesn't cost any crowns, nor does it take a membership.