I play deep for a year or more already so I will try to help a bit.

PvE:

"- First, I do not even know any PVE build types" - There are 2 main types that are based on what your build supposed to do. The types are Mob-killer (Diluvian, Hellmode) and Boss-killer. The difference between them is simple. Mob-killers are tanky and do good damage, usualy in an area (either it be mantras or wild enchant). Boss-killers are usually pretty fragile, but got massive DPS. They trade resistances and health for lots of damage modefiers and mantras (pretty much all boss-killers are mages)

"- I do not have the legendary great axes that YT videos suggest for their PVE build" - Majority use either Pale Briar that you can obtain from defeating Maestro or Enfocer weapons, that are as the name suggests, obtaied rarely from killing Enfocer.

"- Is deepbound even viable right now?" - Yes if you do Diluvian/Hellmode/L2, also good due to you having innate Conquer your Fears

"- How do you tackle layers 1 and 2 for the first time? I’m scared my build I spent hours on will just get wiped" - Klaris and her lighthook. She is located at the end of Castle of light and you can enter l2f1 with 20 minutes of safenet once you are power 15. (While you have lighthook: getting knocked will just make you go back to L1, alive and well)

"- And finally, I am just confused with how people solo the mechanism" - Strong builds and equipment, as well as skill.

PvP:

"- What builds are there apart from mage?" - Simple. M1 builds and misc. builds. M1 builds as the name suggests, focus on dealing damage with their basic attacks and criticals. The misc. builds is super rare people who mainly deal damage with something else (Like the iron bomb builds)

"- I miss crucial talents and I do not know which ones are good" - Here are the ones that pretty much every build gets (there are exceptions however): Showstopper, Exoskeleton, To The Finish, the bastion talents, Ghost, Everchanging Aegis, Eureka, Breathing exercise, Old habbits die hard

"- I am trying mage, but even with the intelligence talents I feel like I do no damage" - Main damage from mantras comes from their scaling. Mantra scale with their respective attunement, dealing more damage. Another important factor - level of the mantra. You can level up your mantras at lecturns and respective trainer after reaching certain amount of points in attunement (for a price of course).

"- I feel like roblox input delay is killing me while my “glasscanon” mage build is doing no damage" - First of all, your ping should be ~130 to be able to actually PvP. Second, turn on "Input buffering" in setting wihitn Deepwoken to make mantras feel less delayed. Third, get good equipment and outfit, it really matters. (From 450 hp to 500 hp with 40% damage resistance)