A couple points of clarification:

If Shifting Woodland becomes a copy of a room, initially both sides will be locked, however it will remember which sides you unlock. So if you have it copy Moldering Gym / Weight Room and unlock the Gym to find a land, then next turn use the Delirium ability to copy Walk-In Closet / Forgotten Cellar you will be able to play lands from your graveyard, as you already unlocked the left room, even though it was a different room when you unlocked it.

709.5c "Left half unlocked" and "right half unlocked" are designations that a permanent on the battlefield can have. Together, they are called the unlocked designations. A particular half of a permanent is said to be "unlocked" if it has the appropriate unlocked designation. Otherwise, that half is said to be "locked."

Shifting Woodland does in fact keep the levels, but does not remember what they are. So if you make it a copy of Scavenger's Talent and level it up to level 2, it will have ability for the remainder of the turn, but lose it once it stops being a copy. If next turn you copy Druid Class you'll be able to play an additional land as it's already level 2. If you instead turn it into Warlock Class you will not be able to use it's level 2 ability as it is already level 2 so there's never a time it becomes level 2. You can still level it to level 3 in either case.

716.2b A level is a designation that any permanent can have. A Class retains its level even if it stops being a Class. Levels are not a copiable characteristic.

While the reminder text for Saga's says "after your draw step," it specifically adds the counter during your precombat main phase. This means after you draw for turn and see your new card, you can still activate Shifting Woodland to copy a saga and get the 1st counter to trigger that ability.

505.4. Second, if the active player controls one or more Saga enchantments and it's the active player's precombat main phase, the active player puts a lore counter on each Saga they control. (See rule 714, "Saga Cards.") This turn-based action doesn’t use the stack.