OS X El Capitan suddenly can't open DMG files

I had the same problem, and managed to open .dmg files by going into Disk Utility>File>Open Disk Image. Once opened in Disk Utility, Double Click (Right Click) the icon and click show in Finder. Then I could install what I wanted! Hop it works for you!

If the Time Machine backup you used to restore the Mac was created after you started noticing problems, maybe whatever drive corruption there was messed up some files that were backed up, leaving you with a good drive but still bad system files. If you have an older Time Machine backup from when the Mac was functioning properly, that might be better to use.

Seeing as you can still basically use the Mac, I would recommend reinstalling the iMac from a fresh install, and then manually moving over your apps and files or using Migration Assistant to transfer files and settings from the MacBook Air to the iMac.