Faster LAN will help you transfer to the machine faster. its an extra cost that is only worth it if you find it takes too long to send files to it. This is your call, if you do, look for Intel's Pro low profile card.
the normal lan(100 megabits/sec. closer to 80 in reality most of the time) will be ok as long as you are not streaming multiple HD streams(ripped blueray 30+ megabit per stream). if you just watch DivX/XviD they only use 0.5-2megabit per second. MCE recordings and DVD's take 4-10megabits per sec.
As i said, i did do the same thing with a slower machine and it worked ok. you should have no problems.
OS is a personal choice. I had a copy of XP Pro laying around so i used that. In my case being able to install other software i use on there worked for me(I also later upgraded it to Media Center Edition and more hard drive space + new cpu). but with linux being as free as it is, i can see many reasons to use it.
What your building will be essentially a NAS(Network attached storage)
I did a quick look for a linux distro with that in mind and found this. May be worth a look.