If I would like to cook something of turkey or pork or anything else and the meat is minced, can I say minced turkey or minced pork? Or I can only use minced beef for every kind of meat?
You can have 'minced' any meat.
We would say "ground pork" or "ground beef" in the US.
What do you say for chicken or other poultry in the US?
If it's ground up, it would be "ground turkey" or "ground chicken."
The only references to "mincemeat" here are either to old fashioned Christmases with mincemeat pie, or a threat to do bodily harm and "turn you into mincemeat."