The 22" monitors usually only carry DVI and VGA. Bump it up to 24" and you'll start seeing HDMI right away even in the cheaper models.
I entrust you in the hands of the good folks at http://www.prad.de
for monitor basics and advanced knowledge. They've got comprehensive articles and reviews that should point out the different aspects relevant to choosing a monitor. If you want a really short summary, mine would be: input lag and ms are relevant for gaming, viewing angle and stand for multimedia use and colour accuracy for graphical work. Go and see the monitor's picture before you buy, if you can, as it's all about what pleases YOUR eye. You WILL adapt to any monitor you choose.
Regarding remote desktop/RDP: I've only used VNC variants and Teamviewer, and those have worked exactly the same over a second remote connection. So yes, I think it could work, but I can't verify it.
I would start my search from this model here: http://www.tftcentral.co.uk/reviews/benq_gw2450hm.htm