They're not related. You could look up their etymologies on this site to find out.
Your second question you should ask on the Name Opinion Board. I wouldn't use them because I think meanings are important and I wouldn't want to have two kids with names that are essentially the same. But they don't sound alike, so if that doesn't bother you it would be fine. Most people aren't going to know that Berenice and Veronica come from the same place.