Can Double Zipping Cause Problems?

Zipping zips is fine. If there is a problem, that would be a bug in the compression software :).

You will probably not get any further compression for those files though unless you're using a higher level of compression.

Zipping a zip might confuse AV software. There are zips, which contain themselves as a new zip of themselves, which leads to an endless recursion if your AV software tries to inspect a zip in a zip, and to avoid such an recursion, the AV software might make a false alarm.

Tools which search in zips like zgrep will not inspect zips in zips. But if you are aware of it, this might not be such a problem.

The compression result can suffer. Think of 2 zipfiles:

a.zip
- a.txt 
- b.txt 

b.zip
- b.txt

a.zip and b.zip can be very different and hard to compress further. If zipping

combined.zip
-a/a.txt
-a/b.txt
-b/b.txt

both b.txt-files can be compressed very good.

So deflating internal zips before compressing the whole thing might lead to smaller results, but more importantly, it leads to better usable results.

You can do that, you shouldn't see any problem but compressing a compressed folder, you won't gain a lot of compression. That's it...