Should be: "Там нема вина." Or:"Нема вина."

These variants should be accepted, too.

The English 'there' often acts as a dummy pronoun, not referring to any place in particular. So, when it's a dummy pronoun, it should not be translated; when it refers to some specific place, we translate it as «там». Since we don't have context in Duolingo, both those options are possible. So, I believe both versions with «там» and without «там» should be accepted.

As for «нема», it's a variant of «немає».

Where did this "є" on the end of a word come into being?

Originally, «немає» was derived from «не має». So, «немає» is the original form, and «нема» is either a reduced form of «немає», or a form changed under the Polish influence (Polish analog of «немає» is 'nie ma').

It's like someone is trying to 'rearrange' the beautiful Ukrainian language into something it isn't.

This is how Ukrainian is spoken in Ukraine. No one is rearraning it, it's spoken this way.

There is - безособове "є" (є вино)
There is no - безособове "немає" (немає вина)

Там немає вина - There's no wine in there
Тут немає вина - There's no wine in here

є що? пиво, де? у моїй чашці :)

It's not that typical, however, to drink beer from чашка. Келих would be more appropriate for that