MapleStory Chaos: Profession Recipe Guide

Proof of Investment - Can anyone clarify how they are gotten from PvP? Going by the name, it seems you don't have to win to get them, just show "effort"?

So many different materials... do we get an extra inventory tab for profession items?

When you play in PvP, you have a score on the side of your screen. Every 10 Score gives you 1 Battle Point (BP). Then you can trade in the BP for Proofs at a ticket counter.
500 BP = 1 Proof
1500 BP = 4 Proofs
2500 BP = 7 Proofs

You get a 4 slot bag for either mining or herbalism (whichever one you choose) after you're done the tutorial. You can get different bags through the event quest when it first comes out, raising an Item Pot, and from various events.

So aside of being able to create all sorts of new materials, dissasembling and making awesome (and expensive) potions, Alchemists can fuse equipment too, right? I heard this gives items potential, but I don't know if it's true and if there would be any other reason to fuse equipment. So could someone please explain this to me?

Fusing equipment re-randomizes the equipment, giving it new stats and the potential to be unidentified. You have to fuse the same items for it to work. Its a really low chance to be unidentified though.

I think I am wondering about regular jewel forging. It looks like you can create plates and jewels with less ore if you use your profession skill. This would tell me that I should not bother to refine my ores and just keep them in Ore format so that I may get more value out of them. Am I correct?

Yep.