@Videobroker, yes, you can legally adopt your girlfriend's child:
According to that link it can take around 2 years.
if I'm married to the child's mother, that doesn't automatically make me legally responsible for the child, not until a formal adoption has taken place.
Yes, correct, but it would speed up the process.
What is the cost
Sorry, no idea. If your girlfriend is Polish, can't she look for such information herself? Unless you're looking for foreigners with similar experiences on this forum?