The Hyundai Veloster has had so-so reliability scores, and few if any people have raved about its handling. Great styling, though. Same engine and related chassis in the Kia Forte SX, so that might be another option.
The Genesis Coupe also hasn't set the world on fire, but it's an inexpensive way to get a powerful rear-wheel-drive grand tourer. I'd get a 2013 or later, as the update that year significantly refined the car. Make sure the ride isn't too firm for the driving you'll be doing.
With the Mazda3, I'd get the s with the larger engine and 18-inch wheels, not the i. If you're getting a manual transmission, one was offered with the s from 2015 on. The Mazda is probably your safest bet in terms of reliability.
Other options include the Ford Focus ST (or maybe even the Focus SE with a manual transmission, avoid the DCT) and Volkswagen GTI. I don't know how well you'd fit in these, but a quick trip to the dealers would answer this question. I much prefer the 2015 GTI to earlier ones, but there probably aren't any of these in your price range just yet. An older GTI would probably be the riskiest of any of the cars mentioned, but most are okay.
I find the VW roomier than the Ford, but I'm only 5-9. As with the Genesis Coupe, the Focus ST might ride too firmly depending on the roads you drive. Some people also find its optional Recaro seats overly tight.
In sum: safest bet is probably the Mazda, while the most fun would be the Genesis Coupe or the Focus ST.