1. Is it possible to convert AP into Diamonds?
2. Can I benefit from having Proficiencies in multiple elemental trees? For example, could I get a huge amount of Health by putting points into all the Hearts (10/10 wind, 10/10 earth, 10/10 fire, etc.)?
3. What benefit do I get from stat breeding? Don't all pets level-up the same?
4. "A second wave of Bandits are raiding Esior's Haven!"
Where exactly do I go to kill bandits?

Also, when capturing pets with nets, does lowering the enemy health improve the success chance or is there no effect?

1. No.
2. Yes.
3. It's not too significant at first but breeding to get better base stats can help get pets that are better for doing certain things.
For Example, Zolnixi's base stats:
Health: 100
Mana: 80
Strength: 10
Intelligence: 12
Dexterity: 11
Agility: 12

But in breeding to get better stats you can end up with something like this
(Taken from a random lvl 0 Zolnixi I have)
Health: 213
Mana: 93
Strength: 23
Intelligence: 24
Dexterity: 20
Agility: 23

That's nearly double the stat. If the above had been continued in being bred for stats it could reach an even higher number which would be useful in those harder battles.

4. The mini events/raids are located in whatever area that's listed. Esior's Haven means you go to the town of Esior's Haven. You won't be able to actually get into the town until all the bandits are gone.
If it gives another location like the Sanctuary of Serielle then that's where you have to go.

5/extra. Some believe that it does improve chance while others do not. You'll have to get more information on this one from someone else as I don't know. I just know I seem to have a better success rate on days where I've lowered the health and other days it just seems to make them more likely to run so I cannot be sure.

6. Has anyone made an in-depth breeding guide to improve stats? I've heard starter pets have good stats, so it'll probably be good to start with them.
7. Does equipment help with Missions or are they entirely based on 'naked' stats?

6. Not in concrete, but there are words of advice here and there about it.
I would PM Oogbar or Eimell if you're interested. I know the former has been working on stats specifically for a long while, and the latter was in the middle of making a guide.

7. Equipment helps. The final number that appears on the stat of your pet after all point allocations and equipment is what missions will account for.

8. If breeding is based off of base stats and base stats increase by level, then is it more beneficial to breed high level pets?
9. Are there any ways to get diamonds without using gold or real money?
10. How do I list my pets as NOT for sale? If I leave it at 0, it says anyone can buy them. However, the stupid thing won't change to my ID when I edit it in place of the 0.

8. I don't know the answer to this but I'll test this with my original starter and mate because I know their original base stats, but based on a previous breeding I did with another of that species I'd say no.

9. You can trade things people want to get diamonds. These include highly wanted event/wheel pets like the recent SH Luffox and avatar items. Those might be short at hand at getting at this time though because the event just ended. Essences, mostly enhanced and mythical, also could be sold for a set amount but they're hard to get so I recommend lots of missions and lots of hatchery clicking if you're going to take this route. There may be other ways but at this time I do not know.

10. Your pets are automatically not for sale. Pets cannot be sold for less than 1 gp so setting it to 0 clears any pricetag. As of current, none of your pets are for sale if that's what you're worried about.

11. How many Philters can I use per egg at one time? Is it only one Philter period or one of each type?

11. One philter per egg. the philter can only be used when it is hatching. if you chose help the pet will get a health boost, if you let it hatch it will be a random stat and if you use a stat philter it will boost that specific Stat.

12. So Philters are worthless? I get a +5 stat increase with each hatching even if I don't use Philters?