as charper1 stated, this would be your cleanest install, however its a moot point, if your current setup is working that is, because the tv is going to downconvert the hdmi signal to dvi if you use that cable. so better pic quality, well in my opinion no, i think it would be the same, other opinions may vary. I may however be able to solve your remote issue, im assuming you have 2 separate dtv remotes both programmed to the one tv. look in the upper left corner of the remotes and grab the one that says rc65rx, notice the R before the x, this is what your looking for, mark this remote in whatever manor you choose to correspond to the HR24, then using that remote turn the tv input to the hr24, and do the following...
menu - settings and help - settings - remote control - ir/rf setup - and change it to rf. Once complete this remote should control only the HR24, and theoretically the other remote should control only the HR20 and would be IR. Im nearly posotive that HR20s needed additional kits to use rf remotes.
*I have the new HDGUI already so the button commands may or may not differ to change from ir to rf, but will be similar.