Leagues have closing dates, but you can join them anytime before that, and play your games whenever. There's a matchmaking system for players kind of like in Hearthstone Arena, so that you're paired against players with your same record. If you want to play all your games in a row, you can. If you want to join, play one game, log off and come back 3 days later, you can. There is still some scheduling to be aware of of, but it's in the form of weeks, not hours or days.
There are prizes, they vary between events. You can check them by clicking the event details link, in-game.
Regarding your questions:
1) Ask your LGS. There is no fixed date (open dueling, whenever you want), but each LGS probably knows when the regular league players meet. I recommend asking your LGS directly. Maybe they can refer you to their league players so you could meet up on a different schedule.
2) From what I've read there is a promo for this Ixalan League after 10 completed best-of-1. Everything additional is depending on the store. Some might supply boosters, but I wouldn't rely on it.
3) Open dueling, there is no fixed starting time. Just find a willing opponent and start playing.
From what I gathered from a buddy of mine is that leagues have a specific closing date to join. Our's in the area was last Saturday, and what happens (sure it may be different in your league) is you are issued 3 packs and you create a 30 card deck from those 3 packs. Then you have a week or so to complete a total of 10 games best of 1 with your opponents. Once you complete those 10 games you are granted an additional pack to update your deck with and so one I believe there is a promo granted after completing your 10 matches, but I'm not sure there are multiples.
Interesting. My LGS allows later joining with the option to buy the additional packs per week for each week you missed. So I started last weekend and would be able to start with 4 packs instead of 3, but I opted for 3.