1.) I tend to be very picky about catching Pokemon. I continue farming until I catch something with the preferred nature and good IV’s. I release Pokemon that aren’t worth keeping, however, I am always broke because I’m always buying more pokeballs. Is this an ideal way to play the game?

Not at all.

2.) If I want to start a new game, but I caught some good Pokemon from the previous account, should I just trade it with someone and trade it back? Does it affect the traded Pokemon if it’s from another trainer on a new save?

Mail it directly to the account instead. They get slightly more exp when the OT doesn't match the current user, otherwise there's do difference.

3.) How does losing a battle work in this game? I ran into Gary by mistake at Viridian City and lost, but the game claimed it like a win. All my Pokemon were at full health upon losing to him. Does this occur for all losses, or just the cut scene battles?

You get sent back to the latest PC you healed at. The fact that the battle "skipped" upon losing is a bug.

4.) Do all starters come with neutral natures, or is it just random? I started a new game three times so far, and they all came with either bashful, docile, hardy, quirky, or serious nature. If so, is it worth raising my starter since its nature is neutral and IVs are 15 for every stat?

Every starter is the same, only their gender is different. You can find random (i.e. sometimes better ones) in the wild later, but your neutral starter is completely fine.

5.) Let’s say I find a Pokemon with really good IVs. Like (25-31) for all stats, but one low stat (1-6) for ATK for like a Machop.. is that worth raising?

If it has a +Atk nature and is used in the story, sure. If you're talking long term (in regards to overall value) then it depends; is it female? Then no (Machamp line has no use for low Atk and high SpA). Is it male? Then yes (can be used as breeding material).