Neotrup that's actually clever. Sagas have small text boxes, I never considered that "after your draw step" might be a templating shortcut for "at the beginning of your precombat main phase."

Shifting Woodland becoming a copy of a card doesn't involve it entering the battlefield. It gains the card's copiable characteristics as soon as the ability resolves. With that in mind:

1. Shifting Woodland will become a Nissa of Shadowed Boughs with no loyalty counters on it (as it didn't enter the battlefield), and immediately after the copy ability resolves it will be put into your graveyard as a state-based action. This happens before you gain priority, so you will not be able to activate any abilities.

2. By default, the rooms on a Room card are locked. When you cast one side of the card, that room unlocks as it enters. Since Shifting Woodland isn't copying a spell, it will become a copy of Walk-In Closet / Forgotten Cellar with both rooms locked. You have to pay the cost to unlock each room you want to use. See the Oracle ruling: If a Room enters from any zone other than the stack, it will enter with both halves locked. (This isn't entering that way, but it's also not entering at all.)

3. Yes, you can level up a Class card that Shifting Woodland becomes a copy of. It won't keep the levels once it stops being a copy, but it does gain all the abilities of Scavenger's Talent including the ability to level up for the turn.

4. No abilities. See the Oracle ruling: If you somehow control Barrensteppe Siege and no choice was made for it (perhaps because another permanent on the battlefield became a copy of it), it has neither of the two abilities.

5. You don't get priority during the untap step, but you do gain priority during your upkeep. At this point, you can activate Shifting Woodland and make it a copy of a Saga like Battle for Bretagard. After the upkeep step is your draw step, which will trigger the Saga as usual. At the end of the turn (during the cleanup step), Shifting Woodland will stop being a copy, but the lore counter will remain—if you do this again, you will get the II ability of Battle for Bretagard, and if you do it a third time it will trigger the III ability and then sacrifice itself.

From what I can see, casting the spell as an Omen is casting an instant or sorcery, so this checks out.