Mostly clear but I still have some questions if you could kindly answer
In the first timeline, did Sayoung die in the apocalypse or before?
So TL2 Sayoung succesfully turned back time creating TL3, but somehow, he alone stayed in his present? If that's how it is(possible theory) maybe it's because time travel wasn't meant to be used by him?
Should we treat this like time travel or multiverse?
Is the watch the thing he got in his awakening? Used to time travel? Also a singular item that is present in all three timelines?
CONTAINS SPOILERS!!!
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1. It was never specified. However, it seems that he died during the apocalypse, since J was PISSED at the apocalypse
2. Correct. On the last part, it was because TL2 Sayoung's powers weren't enough. Turning back time always has a kick back/side effect. The side effect happened to him when he turned back time, making him a warden of the TL2 world.
3. It's more like a multiverse. Every successful turn of time erases the timeline (TL1) and creates a new one.
4. Yes to all
According to Gaeul, there are many fragments of different worlds/AUs, so I considered it a multiverse. But I think you're right, since technically it's one world that rewound time 2 times already.
Memorials can be in those fragments, yes. Although IIRC, it wasn’t specified why the fragment for TL1 memorial dungeon ended up in the Jongno dungeon when they killed the golem (which was why it's Gaeul's first time seeing it too, all she saw were TL2 fragments).

Resending this with additional info
For clarification with what context we have now (no spoilers)
There are 3 timelines:
TL1 - Sayoung never awakened and died, J asked Yeseong to make him a memorial dungeon where J's memories of them living together are used (to cope, because he can't keep saving people anymore if his only success is dead). J turns back time successfully because of the apocalypse.
TL2 - This is the timeline that we see in Gaeul's fragment. J died instead and Sayoung turned back time because of the apocalypse. But for some reason, Sayoung is still there wandering about.
TL3 - This is the current timeline, and the last chance for them to solve everything.
Gaeul meets TL2 Sayoung and helps her not get lost, then disappears. TL3 J tries to save her but ends up getting stuck in TL1 J's memorial dungeon, and TL3 Sayoung gets dragged in too. For the memorial dungeon to end, the day has to end. They both get out but is still technically in a memorial dungeon (TL2). They kissed then they try to find Gaeul. After finding her, Gaeul mentions meeting TL2 Sayoung. Then, TL3 J thinks of the watch and wants to get back to the restaurant to check out its description to find clues.
So they set off to find a way to get out. They find a pile of ash and upon inspecting, they're actually bones of the hunters that J was with in the West Sea Rift (like I said, they're in a memorial dungeon of TL2, so you can make the connection now). J breaks down and Sayoung sees a weird memory of the same scene, then comforts J. Definitely read the novel since it's described better there and hurts more. Then, Sayoung suggests stabbing the Basilisk's skull to get out of the memorial dungeon. Then end of season 2!!
About the Basilisk's Fang, aka the tsundere black sword that J has, it belonged to TL2 Sayoung. That's why it thought TL3 Sayoung was its owner. However, TL3 Sayoung doesn't know of the sword of course.