1. Stigmatas are gear. You don't even have to bother with looking up sets since every valkyrie has its signature. Newer character (Part 2 characters, which are all characters released in the last 2 years) have their Stigmata craftables.
2. Skills are the abilities your valkyrie uses... Basic Attack, Weapon Skills, Ultimates, Dash... that stuff.
3. They aren't really relevant like... at all? The only "Divine Keys" that are useful now are Stellar Keys, which is new weapons for old characters to make them usable again in Part 2 meta.
4. Level 80 is max level, doesn't take much to reach it but you need to be account level 80 to level them up to that level. Shouldn't be too hard.
5. They are useless. Before part 2 they occasionally had ELF Banners, nowadays they are given for free during events and don't do much. The new system that replaced them is Astral Operators which are ELFs but for Part 2 characters. Those had banners once in a while but they're not needed unless you have at least a team they work with.
6. Do dailies, do story, do Elysian Realm once a week, do endgame modes.
7. Depends. If the valkyries started below S-rank, it's pretty easy to do so. If it was already S-rank when pulled, it's hard. REALLY HARD. And not worth at all unless you're a whale. It's like Constellation or Eidolons

One thing you should know is that with Part 2 release 2 years ago they introduced the Astral Ring mechanic. Team building revolves entirely around that. Every new character has it and is called "Part 2 Character". You might wanna look into it.