The English continuous is also often used as a future tense, is that the case with Turkish too? Does "yine geliyoruz" mean "we are coming again [soon, Tuesday, etc.]" or just " we are now in the process of coming again"? If it is also a kind of future tense, is it used for future events which will definitely occur, as is the case with the English continuous, or does it express general possibilities as well?
Yes, just like English, it can denote a future tense.
Actually, it's more flexible than its English counterpart as it can be used for the past tense as well:
"Geçen gün yolda yürüyorum…" (The other day, I'm walking on the street…)
Of course the intended meaning is "I WAS walking on the street," but the present continuous draws the listener into the story, as if it were happening now. It's a common thing in speech; not to be used when writing formally. :)
This can happen in spoken English as well. Maybe not as common as in Turkish, but it's definitely possible to tell a story in the present continuous.
"So last week, I'm hanging out with..." is a pretty normal way to begin telling a story.
Obviously not to be used in formal writing either.