Why are players segregated into divisions?
To make you have a more localized group of people to compare yourself to. It's fairly arbitrary, but moving up within the division is a good motivator to make you want to keep playing.
Are you more likely to be placed with/against players from your division?
Not really. This depends on player availability, and even if division-mates are favored by the matchmaking service, you will very rarely get a chance to play against someone in your own division.
How do you move up within your division?
Winning games can net you anywhere from 2 to 50 points depending on your bonus pool and the strength of your opponent. Your rank in your division is based on the number of points everyone has. Losing matches makes you lose points and move down in the division in a similar way.
Can you move from one division to another? Can I choose one?
The only way to change divisions is by qualifying for another league, either moving up or down. You will be automatically assigned; you cannot choose your division.
How many divisions are there, and/or how many players can be in one division?
There can be up to 100 players in any division. The game will create as many divisions as it needs to fit everybody who played their placement matches into one.
Do the divisions really mean anything? Are the lower/higher rated divisions better?
Nope. The divisions are largely irrelevant.
How do you view other players in your division?
If you go to your leagues screen, you can click on the league you want to view everyone in your division. You can also view other players' divisions from their profiles.