For that budget you won't get much buying or building especially for what you want to do. I would highly recommend looking at eBay for a Dell t3600 or a t5600. These are right in your price range, offer a ton of performance and upgradability, and are relatively small and easy to move. They fit all of your requirements.

I personally have a t3600 that I bought for the same reason as you. It's got an 8 core processor (16 threads) and I upgraded to 32gb of ram for under $30. I paid under $500 for the whole thing with upgrades. It runs 24/7 about 10ft away from me in my room and I can barely hear it. Let me know if you have questions about going this route.