To answer your questions directly:
You can only create a character starting on Normal difficulty. You'll have to defeat all 4 act bosses and complete the Prime Evil quest on Normal difficulty to go to Nightmare and play with your friend who is on Nightmare, and you'll have to redo this process for any character you create. This follows for other difficulties as well. Edit: There are now level restrictions: you must be 25 to advance to Nightmare, 50 to advance to Hell, and 60 to advance to Inferno.
You must be online when playing Diablo 3, so you can join your friends whenever you want. You can join a friend's game without invitation unless it is private. If it is private, message them asking for an invite.
If your friend just buys the game you can join him in his quests, but he cannot join you until he has unlocked the difficulty you are playing on.
One important thing to note is that the only restriction for friend joining you is that they have unlocked the difficulty you are playing on. Unlocking quests doesn't matter; only unlocking the difficulty does.
So if you want your friend to play with you, change your quest to a quest on their difficulty. Note that this doesn't mess up your save file or anything; you will be able to return to your previous quest and difficulty after you've played with your friend.
First click the Change Quest button in the main menu (notice I am currently on Hell difficulty):
Then select Normal Difficulty from the drop list:
And finally select the any quest (here I select Prime Evil) and click the Select Quest button:
Note the warning: the only drawback to this is you lose all progress for your current quest. So finish up what you were doing before switching your quests!
After clicking OK, then you should notice your current quest change in the main menu:
And now when you start the game your friend can join you.
I've done this with my friend to skip the Butcher on Hell difficulty, mostly because I wasn't geared to fight him toe-to-toe. Proof is here that joining your friends can help you skip bosses (I skipped Butcher but killed Belial):
However, to advance to the next difficulty you will have to defeat all four act bosses.
Many people are discovering this problem and very clearly the easiest method is going to be to try to work with people who are on the same quest as you.
Given that will not always be possible the choice comes down to helping your friend catch up or speeding your friend to your quests and allowing them to replay them in their own time.
By maintaining a large friend list by joining a clan, playgroup or adding your name to the spreadsheet of a forum you participate in, you should be able to find someone who is in the same general vicinity as you. Unfortunately there is no method at present to group friends or to edit their names into something more familiar to you (eg a forum ID that you recognize instead of their BattleTag/RealID) so you should maintain a list somewhere to help you recall.
A great number of people (myself included) are deciding to solo Normal mode so that they can take all the time in the world to experience all the details and nuances before worrying about co-op play.
With more experience I've also taken the following action; play one character as my "get through the game, unlock things and get good loot" character and started another that I quickly levelled up. In my case my primary is a demon hunter and my secondary a barbarian. I get to play different content but still get to keep pace with my friend.
I also installed skype to give us voice chat, which helped a lot.
Best Answer
I believe anyone in a multiplayer game can skip dialog or cutscenes, and the skip effects everyone in the game. Sometimes it's just enough to say "hey, I haven't played this yet, do you mind?" - but at other times you might want to just go solo for your first playthrough.
If, however, you're committed to playing with others, and those others are being unreasonable-poopy-head-dialog-skippers, you can review the quest dialog and cinematics at your own pace whenever you wish.
In addition to using the main menu cinematics option (as has been already suggested) to see the in-game cinematics that occur between chapters and at major plot milestons, there is a good database of all the dialog in the game, for each class, on the Diablo 3 DB pages.
For example, this link will show you the quest dialog for the very first quest in the game.
I tend to play Diablo 3 via a "non-Steam game" shortcut, which gives me the Steam overlay. With the overlay's browser, you can be on the DB while you game.