Превод - Руски-Английски - Skoro, mi begali s zurabom tolko ...

Hi Lilly, It seems ok and it conveys the meaning of the original. But I would change "we were running" for "we have been running" and it should be "Zurab", not "Surab": "Soon, we have been running with Zurab, we have just come. And Maks is walking with mother (now)".
What do you think? And what is weird with the tenses?

CC: lilian canale

That "soon" at the beginning is what sounds weird.
If we were reporting something that happened in the past, the verb tenses should be different (were running/had just come/was walking), but that "soon" makes us think we are talking about the future and in that case it should be:
"we will be running/have just come/ is walking"

See what I mean?

Yes, you are right
But this "soon" is a seperate sentence. It is not connected with the continuation. Also all the phrases seem to be, for example, answers for the previouse questions in a conversation, like:
A: When will you come? (or any question with a possible answer "soon" ). What were you doing?
B: Soon. We were running with Zurab. We have just came (home, for example).
A: And where is Makas?
B: He is walking with mother(now).

Could I explain it a bit?

CC: lilian canale

OK, then I think we should punctuate that differently. Do you think it's correct now?

Yes Lilly, I think it's perfect now!

Thanks for your time, Sunny

I will accept the translation then.