Are there any known compatibility issues with using these programs together?

I am using free versions of AdwCleaner, Avast Anti-Virus, Malwarebytes, Spyware, eBlaster and ZoneAlarm Security.

I've not seen any problems, but just wondering     

I don't know of compatibility issues, but your system resources is taking a needless beating with all those programs running in the background. Avast AV and Malwarebytes should be more than sufficient.

I only run Defender (which is preinstalled with Windows 10) and the premium version of Malwarebytes. My laptop, which runs Windows 7, only has MS Security Essentials and the premium version of Malwarebytes.

To add a bit to scuzzy's advice, I would recommend you dump eBlaster and ZoneAlarm.  I am using Win7, I only run MSSE in realtime.  I use the Windows firewall. I run Antimalwarebytes every couple of weeks, AdwCleaner only if I have a problem, ditto with RogueKiller.
I really think for the typical home user, the whole malware thing is way overblown.

You didn't mention which version of Windows you are running, but do yourself a favor and get rid of the bloat. More important is that you use common sense in the sites you visit and the things you download/install, as we are often our own worst enemies.

If you are using Windows 7, then MS Security Essentials (MSSE) and the built-in Windows firewall is all you need. If you are running Windows 10, then the built-in Windows Defender and firewall is all you need. Having said that, I appreciate the extra protection provided by Malwarebytes Premium and I believe it is a good investment.

Looking back at your poast, I'm guessing that "Spyware, eBlaster" is actually SpywareBlaster. The free version does not use up system resources, so there's no harm in installing it. SpywareBlaster does not conflict with any other programs. The free version requires manual updates and tweaks your browser(s) by adding restricted sites. It's merely a prevention tool and does nothing to clean up potential problems you may encounter. The only benefit of the paid version is that it updates itself automatically.

I have Windows 7 still (Windows 10 messed up my audio the several times I tried it)

Yes, SpywareBlaster is the right name

I recently used AdwCleaner to remove a relentless pop-up ad for "Ad Remover" that nothing else seemed to recognize

You guys remain my first go-to forum for computer help. Many thanks