1) Unlike Diablo 2, in D3 you select the quest you want to find a game for, click a button, and the game will automatically match you in a public game with other players starting that quest. You don't have to hunt through a list of games to find what you're looking for. If you're looking to play with a friend, either create a party with them (through the friends menu) before starting the game, or if they're already playing when you log in just click their name on your friends list and (if they have set their privacy options correctly and if there is room in their game) you can just click a button to join their game directly.
2) Yes, they will function similarly to those in Starcraft 2.
3) Not exactly. In D2 you'd use the F-keys to switch between two active skills; in D3 you have a hotbar with up to 6 active skills - one for LMB, one for RMB, and four others. You can bind whatever hotkeys you want for the four other skills through the options menu.