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.
As of a recent beta patch, runes are no longer items that are dropped. Instead they are basically skill modifiers that are unlocked as you level up. Each time you level up, you will unlock new skills and/or rune choices that affect your existing skills. You choose a certain number of skills, and a corresponding rune for each, to equip as your skill set. Each skill can have up to 6 different rune versions of it, but you must select which one you want to have equipped at any time. There are no longer different levels of rune abilities, it is simply a matter of having it unlocked or not.
Best Answer
After some experimentation, I've noticed the following:
Monsters do indeed level up with you. You can see what level the monsters are currently by looking at the top right of the screen:
The easiest way to tell monsters are gaining in power is by turning your difficulty up to something like Torment and looking at their max health (assuming you have health bars and numbers enabled).
Any monsters you were fighting when you leveled do not appear to level. I've escaped to town, hung out in town for a long while, and popped back to see the monsters I've fled from at the exact same HP/Max HP I left them. However, when I started engaging new monsters that were off the screen, I observed the new creatures had much more health than previous mobs of their type. This behavior is different from when another player joins, where after several seconds all monsters gain a proportionally larger amount of health.
Monster health starts scaling up pretty fast, especially once you reach the 50s (and presumably in the mid-to-upper 60s). Thus, the difficulty gets harder as you are quickly gaining levels faster than you can acquire all-new gear and the bad guys are becoming harder to kill. The massive health/damage bonus of Master and Torment difficulties amplifies this effect.
Monsters are the same level as the game's creator and level up with them. Monsters never de-level. For example, if a 23 Monk joins a game started by a 30 Barbarian, the monsters will be level 30, even if the Barbarian leaves. They will remain level 30 until the Monk hits 31, and then will start gaining levels along with the Monk.