Can I install an older OS X over a newer OS X without restoring from backup? Previously I was running Yosemite.  I upgraded to El Capitan and started having problems with an app I used regularly. The backup I have is OS X Lion.  If I restore this backup, I'll lose a ton of files. Is it possible to simply install Yosemite over El Capitan without restoring files from backup?
No - the installer will refuse to run unless you follow the erase and install procedure.


https://support.apple.com/en-us/HT204904 - go to the Erase your drive and install OS X section at the bottom of the article.


You would have much better luck with:


Back up the Mac - make a bootable clone if you like that flexibility or Time Machine is fine.
Install the older OS on to an external drive. Make a junk/admin user when the setup assistant runs.
Add a new user and point that user's home folder at your "current" /Volume/whatever/Users/shortname folder.
Test things that are likely to break: Mail, Contacts, Photos - things that are stored in a database are likely to not be happy going back to older file structures. Once you have a hit list of the things that do and do not break, you can decide to clean things up or manually migrate things to a second "clean installation"


If you don't like the complexity or are not comfortable with the concepts above, you might want to just start with the old OS and move things over one by one when you need them. Having a bootable backup here is key since you may have to boot to the newer OS and export things with a tool or by hand for data that doesn't migrate easily.

If you're comfortable managing several backups and several OS and several home folders, you may never go back to having your OS run on the same volume as your user home folder. I know people (mostly NEXT era Apple employees and others in the profession) that use this method all the time to make OS changes less intrusive and to separate the backup of user data from the backup of system data and caches.