The later integrated chipsets should be ok, as long as they will drive your monitor/tv at the desired resolution. I have a perfectly good nvidia socket 939 based HTPC set up on an nvidia 6850 board, except that the stupid vid chipset maxes out at 1600x1200 or something like that so I can't get 1/1 pixel ratio on my 1080p tv. I had to find a resolution that the tv would scale up without ruining the image, and run it like that. It still looks WAAAAY better than even my old upscaling standalone dvd player because nvidia put a lot of effort into dvd scaling and dvd playback image quality, but it's maddening that I'll have to buy a separate vid card if I want to drive the tv at it's native resolution.
But I think the 7xxx series and later integrated chipsets should work fine with even a 1080p tv at native resolution. Just make sure you check the specs on the mobo and chipset manufacturer website before you buy lower-end, otherwise you might end up having to buy another vid card anyhow.
As for sticking with integrated video in the first place, it will lower power draw and keep the system cooler, which reduces cooling load so you can keep the fan speeds down. That makes it quieter, always a good thing. Other things to consider are finding a really quiet PSU and hard drive, or go with a tiny SSD boot drive and then use LAN attached storage so you can keep the noisy hard drives in another room entirely.
You probably want DVI or HDMI output, so make sure the mobo has that if you want to use the integrated graphics. Also, make sure the mobo has the right sound output. The mobo I have was purchased in part because it had the right digital SPDIF output to go to my stereo, plus a riser card with optical sound out. That was an important choice.
My HTPC uses that case but in aluminum, not black. I wish I'd gotten the black instead. It was fairly easy to work in and standard PC components fit fine, but it's bigger than most stereo components so make sure you have room wherever you plan on putting it.
Consider one of the WD "green" drives. They ought to be quiet and cool, and you don't need blistering speed in an HTPC.
If you buy a new vid capture card, make sure it's digital tv capable since analog tv is on it's way out.
That cpu ought to be ok, but you WILL be going to some sort of hi-def playback at some point, so plan ahead with either cpu speed or upgradable video card that can decode the vid stream. I think you'd probably be ok with blu-ray decode on that cpu as-is, but you'd have problems transcoding on the fly or doing anything else during playback.
Remotes - I got a cheap OEM windows MCE remote and it seems to work fine both under windows and under various myth-tv based distributions.
For software, I'd just try out the various linux based htpc setups. Some are automated enough to detect standard hardware and just get up and running, and then you can really customize it as you learn more about linux. If you want an 80% solution without any hassle, just use some variant of windows. I think the windows MCE front-end is dumb, but if it works then it can make it look like you intended it to look that way.
