Your processing service most likely won't even know what the expiry date on your film is, so it is unlikely to matter to them.
The exception would be really old film that is so old that it doesn't even use the current processes. An example would be very, very old colour film that uses the long discontinued C-22 process.
As for storage, even traditional black and white film will benefit from storage in the refrigerator. Higher speed films need this more than slower speed films. I treat colour films even more carefully.
I tend to make storage decisions about the C41 black and white films as if they are colour films, but I don't really know whether this is justified. It certainly doesn't hurt though (assuming you have room in your refrigerator).