I haven't really seen this before.. The pronoun coming after the verb. Is this a common thing to do and say?
Actually, the meaning is slightly different. This inversion makes the sentence into more of a suggestion / invitation / request for you to wash the pig, rather than a question about whether you are in the process of washing the animal or not.
"Will you wash the pig?" Then?
no, ma a volte si fa.
I think is not common, but it can be said in that way.
Is this some kind of romanian expression with a hidden meaning like i.e "this is not my cup of tea"?
No. As said by Wyktor it is a request, something like "can you wash the pig?". Yes it is a strange / uncommon sentence...