Good luck on your project. To answer your questions:
We had a media room on our first floor. Our teacher didn't like our floor plan so my partner is deciding to put it in the basement. In the basement there really wouldn't be room for an egress (I think this is the proper term) so does it still count as a room just because it is finished?
ANSWER: Yes. It counts as a room. Egress comes into play with bedrooms, where (based on local zoning codes) most bedrooms need a window with a certain minimum dimension and a closet to be counted as a bedroom. But there's no such requirement for any other room, such as a media room, rec room, laundry room, etc.
Another thing, if you have two half bathrooms with another room with a tub/shower in it, do these rooms now count as full bathrooms. I don't think so but my partner for some reason. And we have one full bathroom (being in the master suite), we have three other half-bathrooms. Does this add up to a total of 3 bathrooms?
ANSWER: Half baths stay half baths. You don't add them up. A full bath is a toilet, sink, and shower or tub. Even if you add up a bunch of half baths, there's no way to take a shower in them. However, if it's all one big area with a tub/shower and toilet in it (sometimes they're separated for privacy), then that would be a full bath.
Finally this is a matter of opinion. Would you rather have a fireplace in the master suite with a sitting area or have a fireplace in a family room?
ANSWER: Generally, the family room. That way, it could be enjoyed by more people. But perhaps if it's a vacation home with a spectacular bedroom, it'd make sense to put it in the bedroom.