This week and the last have been pretty hectic, so I entered the lost grove late. I've gathered 7 branches and put them into the barracks but everything says 0% as if I've done nothing. Does being defeated restart your progress? (I die after about 5 battles)

No :)
The progress just updates every 5% ^^
(At least it should)

Hmmm ;; it's been at 0% for 2 days now after the 7 branches...
Thanks!

You're going to need a lot more than seven branches to progress in the zone.
To put it into perspective, in stage two, the lowest amount of branches we need to make any of the buildings is 200.
By this logic, if we assume 200 as the required number, you'd need 10 branches for 5% completion.

Suggestion: grind your pets to at least level 61 (not in the Grove, though, or maybe stick right next to the entrance if you want to try for equip grinding)
Another suggestion for better luck in the Grove, if the pets you currently have on-team are your main team, upgrade their equipment, look again at elements.
Your currently selected team is 1 fire (multi-attacker element) and 2 air (single attacker element), aka, no Tank, Survival, or Healer elements, and all statted the same way (both elements lean towards attack and dexterity, not HP or Agi). No Healer and no Tank is a Bad Idea in the Grove, especially if you're not lv65.

Equips, you're probably running too many pre-grove Epics (Grove rares are fine, Grove equips are a different tier to regular equips of the same rarity), and WAY too many lower level- you might want to send high level statted pets on Saerielle's Very Hard missions to gather some lv50 legendary equips (if you are busy, you could send your party pets on them for the exp)

Oh gosh that's a lot. :,) I thought each item brought it up 5% (from what I heard)
As for the pets, I'll use a guide and try stuff out.
Thanks, Scath! :D