I am planning to build my first fixed gear bike over winter break. I have a 22" 1983 Lotus bike with horizontal dropouts and 27" wheels. I am gonna order either the Suzue or the Nashbar flip flop hub in a couple of days.

First of all, how should I determine what size cog to use? I don't think there is one gear ratio that I could really use comfortably on all the streets I ride on around here. Should I err on the side of uphill and expect to use the brakes more than usual on downhills?

Also, most of the fixed gear bikes I see have curving up handlebars which I think look pretty cool. Can I make them by cutting off a drop bar, of which I have a couple lying around. What kind of brake levers work with them?

This really should be in the FAQ.

For gearing, you need to work out what will work with your frame, chainrings and cog. Its best to start on a smaller gearing, say 42x16 or similar in gear inches (you may even want to start lower) and build up the strength. If you have hills you need to ride on a regular basis then go lower.

See:
https://www.sheldonbrown.com/fixed/index.html - for some good articles on building fixies.
https://www.63xc.com/gregg/101_1.htm - for a riders guide for newbie fixers.

Don't get a Suzue Jr hub, they're cheap and not very cheerful; quality varies widly. I would go for the Nashbar hub, which seems to be the same as the IRO, NYCB, Bens Bike etc. hubs that people are using. Nice sealed bearing unit that should last a lot longer than the Suzue.

The bars you're refering to are bullhorns/cowhorns. You can buy them (ie Chucks bikes has cheap $10 pairs) or cut drops to make some. Any brake lever should work. Some peeps use: BMX levers, standard road levers or TT levers etc. Go and look at the fixed gear gallery for some ideas (https://www.fixedgeargallery.com).

Thanks for the hub tip. I've read most of the Sheldon Brown stuff and I feel like I'm ready to start building it soon as I have the time!

I'm local and I have some Chuck bullhorns I wouldn't mind getting rid of. They're wrapped with a brake even. Let me know if you're interested.