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.
You will not lose the gear that you have gathered.
You will lose save progress if you start at part 1, you can go back to the quests you already done (In your example 7 to 9).
The way the party system works is that you always enter the game at the level that the host is playing at. If they are further ahead of your character personally you will join them at part 7 (in your example).
When you go back to your own personal game, you can start back at part 1 but it will prompt you to save over your spot in the game. You can continue on from part 9, but you lose out on the story line in between.
What you can do is have your friend join you in your game and help you kill the butcher and jump ahead after that is done.
Edit image:
Best Answer
While in CO-OP.
So you gain more achievements playing as a group than solo. You don't lose nothing. But remember monsters get 75% more health for each player in the party.
About the quest progress.
You'll be able to start all quest beyond 4 and the first quest in ACT III. But You'll need to complete the quests 1, 2 and 3 from ACT III.