Situation 1.
At the cinema.
This is the situation
: Mika is at the cinema with two of his friends and they
have already bought tickets online but... they h
ave got to redeem the tickets from
a self-ticketi
ng machine.
I'm not sure what those machines are called. Perhaps it could be 'a ticket dispenser/ a ticketing machine' - Anyone else bought tickets this way and know the correct term?? 'a' is used before this 'self-ticketing machine' because this is the first time we hear about it/you mention it. From now on in your story, it would be "the self-ticketing machine".
HOWEVER, if there are quite a few of these machines, say, in the theatre lobby, then we would still use "a" because you haven't picked out a specific machine yet.
Mika: Did you book the tickets
for Harry Potter 3
as/like we arranged/discussed/planned?
John: Sure. I thought it
might have been
(already-move to end of sentence) sold out already], you know this movie is so popular/you know how popular this movie is, but I found a few tickets left/ it turned out there were these/a few tickets left.(either expression is fine) I was lucky.
Clair: Well-done, fellow!
sorry, googlegoogle, but this sounds like something out of a 19th century English novel, a 1930's Boys' Own Adventure Stories book, or any of Enid Blyton's Famous Five or Mystery of...books! 
lol
Clair: Great. Anyway, I'd like a drink. What about getting some Coke?
John. OK,(or Right.) but we (still) need to get the tickets from the machine.
You've already named the "self-ticketing machine", so just 'machine' is fine here. If there is only one machine in the lobby, then "the", if more than one, then, 'from a machine'.
Where is the ticket machine?
The sentence is fine grammatically, but not quite what one friend would say to another in this situation. eg
'Can you see it/can you see one (if more than one)?'
Mike:
I'm looking/ I'm trying to see one. Oh, there it is!/ Oh, there's one!
Clair: Where?
Jhon:Oh,
I see it. Beside the ticket desk! Hurry
up. The movie starts
in ten minutes.
Mike:
I'm going!! or I'm getting it ('it' here being used =getting on with doing it, getting the tickets)
and
the three of them
run to the machine and
get the tickets
(from the machine.-omit) This is what I wrote for practice, and I'd like your corrections and suggestions.
Please help.
Articles and prepositions are the most difficult concepts in English grammar for me to understand. Don't let the fact that there seem to be a lot of changes discourage you. THIS WAS EXCELLENT. I have taken the time to show you the way native speakers talk to each other, because it seems you are learning very quickly! I have to go out, but will proof this more thoroughly when I get back, so if there are grammatical errors, juggling between what you have written and my reframing the text...)
I'll also add when we use, sold out, booked up, booked out.