-elle is pretty clearly descended from one of the usual Latin diminutives (also visible in e.g. Spanish tortilla). -ette seems to come from a much rarer one?
There's a fun map on Spanish wikipedia of which diminutive forms are preferred in which parts of Spain: https://es.wikipedia.org/wiki/Archivo:Diminutivos_del_espa%C...
Yes, both are used commonly. In names, there are Isabelle, Gabrielle, Cyrielle, Annabelle, Axelle, and many others. I am not sure there is any rule.
https://en.wikipedia.org/wiki/Gabrielle_(given_name)
Another commenter believes, in Romance languages, that the "-a" feminine gendered suffix is diminutive compared to "-o" masculine or neuter suffixes, but that is not true. Not for Latin, or Spanish or Italian or even Greek.