All North American cat associations (CFA, ACFA, TICA) have separate household pet cats classes. CFA will NOT allow declawed cats to be shown in any catagory (including HHP's), the others will allow but don't encourage it.
You can have pedigree (pet quality, not show quality) cats in there; tho a lot of judges will put non-pedigrees over them.
ACFA and TICA have special awards at the end of the year. HHP's in these associations have the same status - you can work on "champion/grand champion" by points for wins. The HHP's are quite competitive too - you can't just pick up a street cat and put them in a show - top HHP's are groomed and treated (sometimes better) then pedigrees.
The only requirement in all the associations is that any HHP over 8 months of age has to be neutered/spayed. A lot of people start out showing HHP's (I did) before moving into purebreds.