Yes! Forget what your teacher told you about double negatives in English. Double (or more!) negatives are totally trending in Italian ;) It is possible to say mai bevo tè con limone but it isn't so common.

There's a great example of this Italian mindset in this video (https://www.youtube.com/watch?v=Fv3fkcCrn6k). He says "I don't have nothing behind my back." The double negative in the English sentence is exactly how Italian (and Spanish from what I've seen) is spoken.

Idioms rarely translate one to one. Come mai? = How come? (Why?)

In this case non bevo mai could be I never drink or I do not ever drink.

It isn't the same thing because lemon tea is not necessarily the same thing as tea with lemon. One could be a herbal infusion, and the other the more standard 'tea' tea, with lemon slice or juice in it (an alternative to milk)

I think that would be tè al limone.

My understanding is that you can usually use either. Sometimes one just sounds "better", especially if the next word starts with a vowel, but they are completely interchangeable.