Your older Mac Pro is severely limited in what video cards can be installed, because it features only 32-bit kernel and will not work with cards that want to work in 64-bit kernel. I do not think any of the cards you mentioned will work in it.
Do not dwell on direct HDMI output -- up to 1080p can be converted from DVI with a really cheap adapter, and from Mini DisplayPort with a fairly cheap adapter.
I think your best bet may be to replace your current card with the Apple-firmware 5770 (one aux connector required) or the Apple-firmware 5870 (two aux connects required). These each have three ports (one Dual-Link DVI and two mini DisplayPort) and will show a picture at all times, where newer cards (if they work at all) will not show a picture until the login screen.
An "actual HDTV" and an HDMI Display may seem different because of the way you use them, but they are indistinguishable.