How fast can you rush difficulties?

Do you have to do any quest to progress or can you just rush Seal of Log?

Equipping Lokarr’s armor, drinking Potion of Clarity like an addict and you will be overleveled without actually doing any sidequest.
Edit: I may have misunderstood your question, I think when you get to Homestead the door is locked until you choose one of the faction, so there is that.

Yes, you have to do a main quest line starting in the Dead Man’s Gulch by talking to Ulgrim. I just can’t tell you whether this quest line has any prerequisites.

Pretty fast if you are skilled and have a sound build. Even the most painful builds can rush if they have twink gear waiting on them in xfer stash for a couple of breakpoints in the game.

Good advice on the Lokar armor and XP pots.

Basically, you can skip any sidequest except the faction choice. I highly recommend you DO NOT skip talking to Kasparov after burial hill quest, the ghost lady in broken hills, or the hidden path. (Hidden path CAN be tough past veteran depending on how fast you are rushing with what build/gear). You won’t permanently lose any of those quests except Kasparov, so you can do most of them later if you choose.

Act 1 is completely skipable by repairing the bridge going west right away.

You also need to kill the Amalgamation to unlock the faction choice I think.

You need to:

Get Ulgrim from Deadman’s gultch to Homestead.
Kill the Amalgamation.
Rescue the Grove farmers.
Kill Karozz.
Do Fort Ikon armoury for the explosives.
Destroy the Necropolis blood wagons
Kill Loghorrean.

I think you are able to walk all the way although much quicker if you have rifts given in multiplayer.