1 Common? Less so than it used to be: most are late middle-aged at best.
2 Full name? Most
Elizabeth people prefer a nn, in my experience.
3
Lizzie?
Lizzie is awful IMO, much worse than stereotypical
Liz.
4 Nn
Liz or
Ellie? Preferably
Liz, but some other nn would be better still.
5 Other nn? I prefer
Betsy or
Libby. My daughter's mn is
Elizabeth and she has sometimes used
Bess, eg when phoning in to a radio programme.
6 Boring? Popular? It is beautiful, but inconveniently long. Popular, yes, but mostly as a mn.