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No
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.
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As a matter of (off topic?) interest, a poster on another site a couple of years ago was having twin daughters, and was really excited when she found that her favourite names, Anne and Grace, were synonyms. She used them for exactly the reason you wouldn't!
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That's just weird!
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