Fit is everything, and height alone isn't going to be enough information. You need to test ride different sizes and brands and have a competant person fit you properly. I'm a little over 5-9, and I ride a 56 Allez. The height was a little tall, but the length is right. I'm sort of short in the legs for my height. You are probably in the ballpark with a 58 or 60 and I'd say a 56 is more than likely too small. It sounds like you are looking for a road bike, but you don't specify, and also don't mention what's available to you locally. There are a lot of good bikes out there at your budget, but does your budget include shoes, shorts, etc. or is that what you plan to spend on just the bike?
Shoe size should be between a 49 or a 50. 50 is actually closer to a 15 but maybe a little large. Depends on how you like your fit, some people like a snug fit, some a little more loose. Again, trying it out will help you decide what's best for you.
"It hurts so good..."