"She is drinking a glass of white wine."
Translation:Ŝi trinkas glason da blanka vino.
Yes, it could also mean that.
You can make a strictly progressive tense in Esperanto but it's not that common - usually, the simple present is used whether we would use simple present or present continuous in English.
(Similarly with the simple past in Esperanto versus pluperfect, simple past, or past continuous in English, often even present perfect.)
Depends on whom you ask :)
Some people use it, some don't.
My impression is that it's very uncommon, and that the sort of people who use it tend to be people who want to play with the language.
People who use the language every day and for whom the novelty of "this cool language that allows you to build your own words" has worn off probably stick to more traditional patterns.
I'd recommend that you not use it. Because it is less common, it will draw attention to the word itself and not to the message you are trying to convey.
A bit, perhaps, as if you said "Drinking is she a glass of white wine"... which is probably grammatically correct English, but unusual word order and so may take a little longer to understand, whereas the more conventional "She is drinking a glass of white wine" will be understood quickly and effortlessly.
You can trinki alcohol as well as drinki it -- you don't have to use drinki just because the object is alcohol.
As PIV puts it, drinki is "Plezurcele trinki alkoholaĵojn en nemodera kvanto" (drink alcoholic drinks for pleasure in immoderate quantities).
So if you drink a moderate quantity, you're just trinki-ing it, not drinki-ing it. And for many people, a glass of wine is a moderate quantity.