Hi, kondora, and welcome to the forum.
It seems to me that the items you have deleted and moved to quarantine were likely to have been malware items (or at least detections) from within the system restore points. Cleaning those items generally leaves that particular restore point unusable. If enough were removed, and from detections within most of your restore points, those will be unusable. It's possible, with experimentation, that you might find a working restore point, but this would be a trial and error thing, and, unless there is a problem with your system, not really worth doing.
Could do with a bit more info, if you can.'
What was the AV installed prior to Avast, and how was this removed?
Was the computer behaving badly before scanning with Avast (eg: showing any symptoms of a virus, or website redirects, or strange unknown icons appearing, that sort of thing?)
How did you remove Avast, and did you have any problem with that?
What AV are you using now? Any other security software?
Apart from system restore failing to restore (I'm guessing it goes through the process, but then you get a message saying "System restore was unable to blah blah...") are there any other problems?
(This system restore issue might turn out not to be a big problem at all.)
Any files you deleted are gone. They might be recoverable with recovery software; without knowing what they were, this is unlikely.
Any files moved to the chest (quarantine) were deleted when you uninstalled Avast. They probably are not recoverable. (If you remember any of the names or paths, that would be good. I'll bet some of the paths had the words "System Volume Information" in them.)
I (and others more experienced) will be able to make suggestions once the answers come.
The suggestions will be along the lines of :
Completely uninstalling whatever AV is (or was) present, and reinstalling Avast.
Running one or more scans and reporting the names of detections.
Possibly installing another anti malware scanner that works well. (Such as MBAM or SAS.)
Removing all old system restore points.
Any other advisable setting changes or tweaks.
We will, of course, provide a step-by-step.