I have ripped my library of DVDs to my home server. I play them over WiFi from other PCs.
Can an Android phone with HDMI out do the same thing as a media center PC, playing these movies over the network on to a TV?
It sounds like you're trying to do something a bit like DLNA? This is the set of technologies/protocols that allow you to play media on one device on another device on a home network. It's designed for connecting things like media servers, media players, TVs, etc all together to play across the home network.
You might find the answers to the following questions useful:
You might consider plex. You would need to install the plex server on some machine and then install the plex player on your Android. Software is free. I've found that it generally works well over wireless. However, I've been playing from my Win7 laptop over to my Roku (with plex installed). Haven't tried it on the phone. In theory, plex works at WAN speeds too; and they provide a login service you can use to find your plex server at home while on the road.
oops... didn't look at the date of the post. Sorry.