I would go to around $7-800 for the right solution. This weekend, I will borrow a friend's laptop for a couple of hours, hook it up to my TV and see whether the flash issues are resolved. It runs Windows 8.1/ i5 4210U.
A slightly left field solution I'm thinking of, is to stick with what I've got for music (quiet, no issues), and go with a combination of IPad mini, Apple TV (for airplay) and Chromecast for streaming video. That takes the whole processor / fan noise debate out of the equation, although I may lose some flexibility down the line and may not be able to cover all my bases as far as video sources are concerned.
I've looked at building my own, but case size cf. Mac mini, NUC etc., is the issue.