It's going to be pretty challenging to pull that off for $500 given your requirements. Since you want gaming, we'll start with a intel i5 quad core....dual core won't be enough.
Intel i5-3470 (3.2/3.6 turbo) is $149 at microcenter - this is a really crazy price, it might not last. The i5-3570k is marginally faster (3.4/3.8), and is fully unlocked for over clocking, but its $189. The 3470 can be OCed a bit, I personally don't think the 3570 is worth the extra $40 in your case. These CPUs are typically $50 more at other retailers.
A non-overclockable h77 motherboard will run you $100ish, an overclockable z77 around $120ish.
8GB of DDR3 is about $60, 4GB is too small, 16GB is overkill for you.
Doesn't sound like you *need* a bluray drive, but if you want to use this box to rip, that's another $50.
Your HDD will work, but WD greens aren't known for their speed. You'll just have to sit at load screens for quite some time with it. If you really want to store a few games at once, you need a minimum of 250GB IMO, I bought a 120GB SSD and I can really only fit like 5 games on it. You have a few options:
1) just use the WD green (free)
2) buy a 32-64GB SSD to use as a cache to speed up things tremendously (~$50ish)
3) buy a hybrid drive (1TB for ~$100ish) - similar performance to the SSD cache but simpler
4) buy a 250GB SSD (~170), highly recommended if you want to splurge, IMO its worth every penny, SSDs will cut your load times down to almost nothing
If you want it quiet, you'll need a quality power supply, a $40 corsair CX430 is an excellent value. You'll also want a decent case....I personally really like the bitfenix prodigy - its a small tower that uses mini-itx boards, but has room for real 3.5" HDDs, and large video cards...and the installed fans are super quiet and cool extremely well ($80 at microcenter). You can cheap out on the case and PSU and get a combo, but what you'll end up with may be big, ugly and loud. (~$50)
Then the final and most important part - the video card. If you want to run bioshock infinite in 1080p, a GTX 650 (~$100ish) will barely run it at 30fps. A 650ti (~$130ish with 1GB) will get you 45ish, a 650 ti boost (~$170ish with 2GB) will get you a pretty solid 60. I recommend Zotac cards, they're super quiet.
So adding it all up, the cheapest way possible:
$50 Cheap case + PSU
$460 before tax and shipping, so probably just about $500. This is of course excluding the OS (~$100ish for Win 7/8), and any peripherals (keyboard, mouse, controllers, etc)
I personally wouldn't skimp on the case, PSU and video card, but its your money.
As far as your other questions:
Most video cards come with dual HDMI outputs, but its a real PITA to manage this in two separate rooms. Between the extra cabling and accessories (extra remote etc)....if all you want to do in the bedroom is stream video, just use a cheap streamer to pull off the net and your NAS and leave the PC out of it. The WDTV boxes tend to work pretty well with local media, will stream a 1080p MKV and netflix just fine on its own and are pretty cheap. ($70ish I think). Likewise, for streaming to iPhone, if your synology can transcode in real time, just use that. If your synology cant, rather than leaving your PC on full time (don't underestimate the impact on your power bill of doing this) as a transcoding video server, you can convert your videos to a more iOS friendly format on the PC and store them on the synology. I've got one of those too, video station works great over the net provided its the right format.
Just to put the power cost into perspective. In order to do what you want, you'll basically need the PC idling 24/7. Burning about 100W idle at 20c a kwh that's like $15 a month. Your power is prob cheaper than mine, but it adds up quick. The synology is always on anyway, and a streamer box will idle at like 5W at most. Over the course of a year it'll basically pay for itself and then some.