Dani for Danica/Danielle - I don't especially like
Dani, as I think it's more of a boys name. It does work as a nickname for those.
Eve for Eva/Evangeline -
Love Eve and
Eva. I think both work as separate names and don't need a full version. I wouldn't use
Eve as a nickname for
Eva since
Eva is so short, it doesn't really need a nickname.
Eve or
Eva both make nice nicknames for
Evangeline.
Penny for
Penelope -
Love it.
Em for
Emma - This is my name, and I always go by
Em, but I definitely prefer to have a longer version to use for official things, at work and such.
Usually I prefer to see a full name used with a nickname, rather than just a nickname on it's own. However there are some which don't sound so nicknamey and can work alone. Your example of
Tessa, for example, works alone without the full name
Teresa. I usually think that very short nicknames (e.g.
Em) or nicknames which end in "y" or "ie" require a full name.
The worst nickname-as-fullname which I've seen is
Hel. Not short for
Helen, just
Hel.