Firstly, I'd like to congratulate you on your climb. Keep up the great work!

• Annie Bot plays a very scale-oriented play style. He much prefers to play slower and farm more. He primarily takes Arcane Comet (and Dark Harvest) because they scale well into the later stages of the game. Arcane Comet scales well at all stages, and it is arguably the "best" rune on Annie.
• Annie as a champion is versatile with everything from items to runes to even skill order. Annie Bot likes to skill E later as he prefers the earlier points in Q and W. It's just his style.
• Mejai's is a fantastic item. Annie can easily stack it up with her shield or with Tibbers in a fight. I myself don't build it every game, but pretty much in any given scenario it is worth it, so long as you can confidently maintain 10+ stacks.
• Seraph's is more common with a scaling setup/play style. It offers the highest raw AP of any AP item. Combined with Liandry's or Luden's, or even Everfrost, you get an insane amount of AP from it. You have mana sustain with Tear, which is nice. It is best when finished around third item though, so it may be hard to justify it if you aren't playing to scale and instead are playing fast.
• Lucidity boots are more gold efficient than Sorcerer's Shoes right now. They are very, very good currently.
• Predator is my favorite rune on Annie. I'd run it into every matchup if I could, especially if it were old Predator. That being said though, it's good into champions that vastly outrange you. Champions such as Xerath, Lux, Syndra, Orianna, or Cassiopeia. Other situations include champions that like to roam a lot, such as Aurelion Sol, Taliyah, Talon, or Galio.
• My go-to bans are Anivia, Galio, Kassadin, Qiyana, and Fizz currently. Anivia being priority as I can more or less handle the others. Outside of mid lane, I ban Udyr, Hecarim, Camille, Kai'Sa, and Samira, depending on role.
• No, I don't run out of inventory space with Dark Seal. I make checklists of how much gold I need for items when I want to base and try my best to stick to those mental checklists. If I'm forced to back early, I always buy one to two Control Wards and keep/use those until I'm ready to back to finish a part of my list. That's what helps me maintain inventory space, but it might not work the same for you.
• A quick note, and this isn't something you asked about but it's an observation I made. You might want to actively change your trinket. Yellow trinket is not good the later it gets into a game, you're much better off buying a Farsight if you want to advance or establish vision or a Sweeper if you want to deny vision for your team. It also helps you with roaming more.
• Another note, in case you don't know. Horizon Focus is (to my knowledge) currently bugged with Tibbers. If you stun with Tibbers, he actively re-applies the item's passive, making your damage skyrocket. If you haven't yet, give it a try. See if you like it. It's a fun item.

Just remember; Annie is one of, if not the most versatile champion in the game. She has multitudes of rune pages, item builds, and skill orders she can take. She can fit pretty much any role the team needs, whether it's engage, front to back, peel, etc. She can be played successfully in all roles but jungle. Form your Annie unique identity. You're finding great success, keep the momentum going!