How acceptable it is to change the tempo when playing a classical piece? I mean, playing some parts with slower tempo and some parts with faster tempo than rest of the song. Is this a good way of "interpreting" a piece? Do professional players do it?

It depends on the era - in Baroque or "Classical" (Mozart, Beethoven, Haydn) music it would be less desirable. Changes of tempo might be used but usually music returns to the original tempo almost immediately. If composers of that era want a tempo change, they'll indicate it - and the more striking the effect.

Of course, no rule without exception - there are a lot of professional interpretations of Baroque music which masterfully play around with tempo.

Romantic music is full of tempo changes.
But IMHO there should always be one dominant tempo which somehow can be felt even if you slow down or accelerate for effect. Else, tempo changes just come across as arbitrary or poorly executed. You have to establish a kind of "standard" or "norm" in order to dramatically leave it, if that makes sense.

BTW: Welcome to the forums! Have fun here!

I think Euphonatrix said it extremely well. It pretty much depends on when the music was written. Baroque, (Bach for instance) is going to have a steady tempo. With Classical period music, Beethoven, Mozart, Haydn - still pretty steady but with a little bit of flexibility. Romantic - tempo can go all over the place.