It should not really be that problem. I am running this config: a 17 inch monitor branched with a VGA kabel and a 23 inch monitor over the same graphics card but branched over the DVI Cable.
If you go into favorits - system settings - monitor and video settings you can determine which of the two is primary monitor and whether to “clone” the primary or to extend the desktop over both. You can also define where it stands (on the right or on the left of the other) and I found out you can very well extend the monitor over both and have different applications running full screen in one or the other. You can easily take an application and transfer it to the other desktop. What I did encounter is that some applications tend always to choose the secondary monitor to open (I did not find anything to prevent this) while some other “remember well” where they have been opened. Some websites when you go on video “full screen” may open full screen on the wrong monitor. You tube doesn’t others do. I think it is related with the type of video stream they use.
The resolution will be set in monitor settings so it is fine, what AFAIK is not possible up to now, is to attribute an activity specificity to a monitor. But I remember that you can change aspect of the wallpaper of the second monitor and it is retained on reboot, without influencing the settings of the first one. I did not try if this is also true for a notebook + ultra base fixed external monitor or notebook and directly branched external monitor.