PvP gear is meant for PvP, as it gives you special bonus meant for PvP, but you could actually use for regular combat, but you'll find it is easier to obtain non-PvP gear that'll be equal or better than the PvP stuff. You only need to be level on the gear to wear it. Just remember not to look at the item level, but the level underneath the item level. The item level can and will later be different, namely higher than the level below. The level below indicates the physical level the job must be to wear it, but item level is what actually affects the stats for the gear. So a lv50, item level 60 (ilvl60) gear will be inferior to a lv50, ilvl70 for the same body part. But both can be worn by lv50.

If the gear on the market board is the same, but better stats it means it is probably HQ (High Quality) gear, so it's stats will be higher. Gear you earn from quests and the NPCs will always be Normal Quality (NQ). So yes you can technically get better gear, but it's not a requirement by any means. In fact, you'll often find gear equal to HQ or possibly better from in dungeons. I made it from 1-50 off almost nearly quest gear, so it's not a requirement. Better gears starts becoming a factor when you hit 50 and the dungeons start getting more tightly tweaked for specific average item level.

Upgrading or getting promoted in the Grand Company is definitely important. If you want to do higher level GC leves you need to be promoted. There is also gear and materials only found from their stores. You also will get a higher seal cap, maxing out currently at 50,000 seals, which will be very useful later when you're not spending seals like mad. So definitely get your promotions.