Kinda new player!

[Teiren]

The gear list preview on your saved class list window only records the gear from the last time you selected that entry. It'll update with a warning that something's not right the next time you select that entry.
Btw, materia really isn't a big deal, so don't expend too much time or money on it. And spiritbond level rises from fighting in that gear, so think of it like EXP for your gear.

Nowadays you likely can reach Level 50 through just Main Scenario Quest and all sidequests because doing all sidequests also means completing each dungeon once. If you feel like you're falling behind, that's what FATEs are for.

About which Summon to use: I recommend you just continue to use Ifrit-Egi from now on until you get more comfortable with the game. The decision between Ifrit-Egi and Garuda-Egi really is personal preference. You can leave Ifrit-Egi on Sic behavior and do totally fine. Garuda-Egi MUST be used with Obey behavior because one of her skills is completely useless if you leave her on Sic, and Obey means she only uses skills when you tell her to by putting her skills on your hotbar.