diablo-3
In Diablo 2 it was very possible to speed run through the acts and end up on a harder difficulty with a lower than anticipated level.
This is quite achievable in Diablo 3 as well, whether bypassing the optional content or being rushed by a friend.
What are the recommended level ranges before progressing to the next difficulty?
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: 
Boss kills are still viable. However the best chance for the best item drops are now off Rare monsters and Champion packs that are found randomly throughout each Act. With the combination of that fact and the Nephalam Valor buff, Blizzard is attempting to provide the best item farming to be systematically clearing areas of Rares and Champions, finishing off with a boss (mainly because you know where they are) as opposed to repeatedly running straight to a boss.
A very good analysis of item farming is available here
Best Answer
Normal Mode should take you from levels 1-30.
Nightmare Mode should bring you from 30-50.
Hell Mode is tuned for level 50 characters and should take you to the level cap of 60.
Inferno Mode is designed for level capped players. You can not access Inferno mode without reaching level 60.