can we use such verbs as "finish, start, begin" in progressive tenses? If yes, could you please some references where such a possibility is indicated, if not, why???

Not normally, because they are 'instantaneous' actions; nevertheless, they do crop up, rarely. Somewhere around here, there's a long thread on starting a car which stretches the imagination.

Dad: There! I've finished the rumpus room!
Mom: Oh, you've been finishing the rumpus room for three weeks now, but you always manage to find something else that needs doing.
Yes, it is possible:
"I was finishing breakfast when the doorbell rang": I was having my last bite of toasted bread when...

"It was starting to rain, so I went into a pub": the first drops of rain were falling...

Students: Are you brave enough to let our tutors analyse your pronunciation?
Thanks, Pieanne-- much better examples than mine.
Teachers: We supply a list of EFL job vacancies
Didn't mean to contradict you or anything, but I think we wrote the post at the same time, it just took me longer to find the examples!