The tear memory thing can be done at any time, but it reveals or hints at things that get revealed over time in the main story. The sky islands usually have treasure chests withmaps in it that lead to chests with clothes and such in the depths, but really they'll usually be around interesting things like the ancient mines and collisiums and GIANT hollowed out tree Stumps.
The depths and dark link armors you get by talking to all of the different Poe statues. There's one at the pools of courage, wisdom, and strength, and a really bigass one in the north west on a long skinny incline. The depths armor is nice, but the dark link one seems to just be aesthetics as it has no gloom protection.
I might get the links Awakening toon link bobble head armor, I was hoping to find it in a chest or get it from the amiibo. Seems it's only available once you scan the amiibo though
Also protip: the difference between Veggie cream soup and cream of vegetables soup is one is made with herbs and one is made with veggies. Useful to know when upgrading your horse to the max.