Home theatre overview compatibility help.

Depending on the physical setup of your room a case that would allow for more air flow so that you can operate without fans would make for a quieter room. This site is very good for this.
http/www.silentpcreview.com/index.php
If you haven't used it yet try out Kodi/XBMC for it.
http/kodi.wiki/view/Main_Page

For that use, an i3 or i5, with a reasonable GPU, is warranted.
Games are a whole different thing than playing movies and music.

if your only use is as a home theater (playing media files, streaming web, etc) and not any sort of gaming then you can get away much cheaper. technically you can make do with an intel i3 or amd apu using the onboard graphics not a graphics card. you can even buy a mini sized case and save there as well and have it take up less space since you wouldnt need a graphics card for just htpc use. since you plan on using a tv you can just use the tv as your monitor as well and save there. the only way i'd say you need a graphics card is if you game or are into heavy tasks which may use one (which would mean you'd want a better cpu than an i3/apu anyways).

i'm not sure why you are buying both the audioengine p4 AND the energy take classics as you could use one system for both pc and tv use (the energy's). provided they are in the same room of course. i'm not sure about the p4's but the a2's are VERY position dependent and can sound good or horrible depending on positioning, how far the walls are away and other factors. you can do better for the money with them. the energy take's however are excellent budget level speakers and are one of the best choices for people looking for good sound on a low budget.

you could likely do a bit better than a sony reciever on that budget. i spent a similar amount (perhaps just a hair more) and use a pioneer which is considered a step up over sony generally. there are of course other good brands too. i'm not saying sony will not work (we have a sony in the other room which for the low price works excellent) just that you may want to at least take a look at other options.

i run a similar setup (you can click the link in my sig if you wanted to see photos) using a pc, tv and home theater. the tv acts as a monitor for me, i use the home theater for my pc, ps3 and any other sources but since i game and run demanding tasks not just htpc use i run some heftier hardware. if you do everything in the same room/area it really does work out nicely.