Depends on the size of the car seat and the check-in desk as to whether it needs to go in oversize - my son's car seat needed to go in oversize at some airports, but at LHR it can go in the normal.

Whilst I can't definitely answer your query, AFAIK you are only allowed to take a car seat for free (i.e. additional to your checked bag allowance) if travelling with a child/infant. Otherwise it will count as one of your items of checked luggage (possibly incurring an extra bag charge)