"My friends stopped drinking."

Because it is the construction with an infinitive, look at the drop down. This happens with other verbs too, so it is worth looking out for. I cannot find a rule that explains why, so it is just another thing to memorize.

The way I understand it is that the friends stopped themselves "from drinking" rather than stopping drinking the way you might stop a car by stepping on the brake or stopping a person to ask then for directions.

Because it means "My friends finished drinking". Not quite the same meaning as "stopped drinking".

If they just stopped drinking, I would think so.

Because that implies immediacy and this sentence does not. Hope that helps

I am guessing in answer to my own question that parar like other "terminating" verbs needs "de" just as initiating verbs need "a" before the infinitives following.

that translates to "my friends stopped to drink"

Often, Duo gives "correct" answers in the feminine but will also accept masculine answers as well, if you're translating to Spanish from English. That's more just to show you that you can do that than that there is some "correct" gender involved. Only when gender is apparent in the English (using "she" or "the women" or something like that) do you have to worry about the difference between amigo/a.

I'm pretty sure that would be "mis amigos pararon para beber."

That is not correct, you need "de" after "dejaron", to say "My friends stopped to drink" you say "Mis amigos se pararon a beber." (Normally refers to when they are in a car and stop by)

In this case you don´t need to think how the sentence would be translated without de, it's just the way it is. In English when you say 'Someone + to be + going + to' you will never consider any other preposition except 'to'. The same thing is with the verb dejar, you just have to remember what preposition with what verb to use.

Some examples you can find here http://users.ipfw.edu/jehle/courses/VRBSPREP.HTM

Not sure what you mean here graahh - dejaron is preterite (past) and your guess dejan is present ,dejarán is the future tens. Use SpanishDict or some other sites for the conjugations. Why no level tags btw?.

No.

It just means to drink in general. Notice how we all assume it's alcohol based on the verbiage of the sentence, but they could've stopped drinking water, milk, soda, anything really. The sentence doesn't actually say alcohol.

Quitar is "to remove".

Terminar means "to finish". I'm not sure that applies here.

Detener is "to stop" in a sense of movement. Like "to stop a car". Or to, quite similarly, "detain".

If it wanted a perfect tense in Spanish, it would give a perfect tense in English.