Building a Media Center (server?) with old parts, need advice

So I have an OLD PC I build like 5.5 years ago that was going to be a conversion project, but was ultimately scrapped. I would like to turn this into a media center possibly with XBMC, but I'd like some advice before starting anything. The parts are below:

• Fractal Design Core 3000
• Intel Core 2 Duo E7200 @ 2.53GHz LGA 775
• Corsair 600W PSU
• GIGABYTE GA-EP35-DS3L
• VisionTek Radeon HD 4850 (might be dying)
• 2GB RAM
• DVD Drive
• 500GB HDD x2
• Windows 7 Home Premium

It probably consumes more power than necessary for what it will be doing, but it's better than just sitting around collecting dust. I think my graphics card is overheating/dying so I'll probably just pull that. I don't need a graphics card for an HTPC, right? Perhaps only during OS setup?

So, if the parts above are OK to go with, what is the best media server software to use? I want something smooth and stable. Also, what is the best way to connect this to a TV? straight HDMI? Through say, a PS3? I would like to watch videos at 1080p, if the above can handle that without a graphics card, but I don't want to have to transcode anything.

Any and all help is appreciated

All of the parts you listed are fine for an htpc.

You'll need a graphics card to display video, since the Intel E7200 nor the motherboard have integrated graphics (nor does the motherboard support it anyways). 

What problems are you having with the HD4850?  The 4850 would be more then sufficient to play anything you throw at it, and it will be able to play a few games too, albeit a medium to low settings at high resolutions. It will have an HDMI port to hook up to your TV and will use the TV's speakers.  Or you can use the DVI port (would need a DVI to HDMI cable) but will not output audio (would need to hook up speakers to the pc instead).

You can use Win7 for an htpc is fine (a clean install would be best), and can install XBMC and Steam on it.  If you plan to watch TV with it using a TV tuner, you can use Windows Media Center to watch and record TV too.  You can install XBMC or OpenElec as standalone as well.

Personally, I have an old Core 2 Quad (Q6600) with Win7 Ultimate that I use, with an HD5450, that can play just about anything (1080p tv streams, blueray, etc.).  I have an HDHomerun Prime TV tuner set up to stream cable tv over my network, replacing my cable box and dvr there.  Installed XBMC to play local media from my home server.  For input, I use a Logitech k400 keyboard with built in touchpad, and if I just want to watch TV I use Sanoxy Windows remote.  Both are cheap and work nicely. 

Though my graphics card is not powerful, I can play a few casual low graphic intensive games too. You can install Steam, and use Big Picture Mode to make yourself a little cheap SteamBox for casual games.

Overall I am happy with my setup.

/ended up longer then I intended, hope it helps.

When playing games my PC would overheat and shut down. I *think* it was my GPU since the temps were climbing over 100C which is huge, IMO. I tried redoing the thermal but it still did it. Since I don't need a huge GPU, I could probably pick something with HDMI for cheap that users less power.

This would purely be fore local media. I was going to install XBMC for the interface, and have it setup so I can handle files via my home network (from my laptop or something). As for input, I completely forgot about that, but I'm sure I could have a wireless mouse on the coffee table or something.

Yea, sounds like an overheating card.  You can replace the cooler with an aftermarket one, but it is already an older card, so spending the money on a more recent model is probably wiser.   I picked up my XFX One (an HD 5450) for $10 bucks on clearance at bestbuy and has played everything I've thrown at it.  It runs at around 15-20 watts, and has passive cooling too (just a large heatsink, no fan), so no worries about fan noise.  I'm sure you can find a similar card, newegg always has deals on those lower tier ones aimed for the htpc market.

Here is the wireless keyboard/mouse that I use:

http://www.amazon.com/Logitech-Wireless-Keyboard-Built-In-Multi-Touch/dp/B005DKZTMG

Sometimes can find it on sale for $20.  Lightweight and works awesome, uses the Logitech universal receiver.

It will work on XBMC and Openelec standalone, no drivers needed.