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.
Each follower has a specific list of equipment he or she is capable of equipping. You can see this list by going to the items page on the official site and clicking the follower's icon at the top; here are quick links:
Regarding the special follower items, those are equipped in their own special slot and do not replace any other item. That slot is not immediately available, it is unlocked later in the game.
Best Answer
Once one of your characters finds a follower, all of your characters have them available so long as you are in the same game mode (i.e. hardcore, seasons, etc.). Same goes for crafters (blacksmith, jeweler, mystic).
As @Arperum mentioned, you should still find a Templar in the cathedral as part of the story line (even if you have Kormac following you, you will temporarily have two followers until after Jondar).
Same goes for the scoundrel and enchantress (bringing along the same followers when meeting them for the second time can provide some funny/unique dialog by the way). You will find a "second" follower of each for their part of the story.
Edit: The followers are unique such that each of your characters will have to gear them differently. I just tested creating a new character and the followers were available in both story and adventure mode (however they had default gear).