What's wrong with my bike?

Seriously.

I love my frame, the cranks and pedals. I'm even a fan of my chain, it's strong and shiny.

BUT, the wheels aren't right. The front's too big and the back's machined.

Not that much of a big deal IMO. If you can afford a new rim, buy that and have the wheel rebuilt, otherwise don't.

The fork's a shitter to change because i can't find a 650c 1" fork with the right length [very short] steerer.

Buy some 650 forks and cut down the steerer.

I don't know what bars to go for, drops? Risers? horns? All have their benefits but will drops be too low? Will horns look too gay? It'll stay brakeless and i don't need it for commuting, just riding for fun.

From the picture I would say drops will be too low. Grab some risers/ bullhorns/ flats from another bike, ride it and see how it feels.