Danièle is the French spelling of the female name. It should have an accent, but in the US those get dropped regularly.
-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...
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.
(1) El isn't a name for God in Hebrew; it's just the ordinary word for any god. Like "god" in English. Conceptually close to ba'al, the ordinary word "lord".
The best-documented confusion of this kind that I know of is the Greek "Adonis" - in Greek, Adonis is a name, but even at the time it was recognized as being a Semitic loan from a word meaning "lord". We know enough about the cult of Adonis that we can identify him with a specific Semitic god, whose name was Tammuz.
(2) You're right about the etymology of these Hebrew-derived names. I think you're on shakier ground when you say that names ending in -elle are "not necessarily diminutive" in French; since the diminutive form exists, people are inevitably going to process those names as diminutives, regardless of the etymology.
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.