The correct term is Household Pets not Household Cats. This class is for cats of no particular breed, obvious purebreds without papers and pedigreed cats that don't fit their standard (Scottish Folds with straight ears, Manx with a full tail, etc) Kittens 4 months to the day before they turn eight months can compete as Household Pet Kittens. 8 months and older, Household Pets. Unfortunately declawed cats CAN be shown in TICA's HHP classes but not in CFA or ACFA.
We can't tell by a photo whether your cat would do well at a show. Judging is based on both how well the cat is groomed and how well it handles being handled by a judge. You could show up with a perfectly groomed cat but have it be absolutely terrified of all the smells and noises of the room and end up hissing at or biting a judge. We also have no clue how well groomed your cat would be. Cats have to be bathed, fully combed out, ears clean, nails trimmed.
See how your cat react to strangers picking her up. Before you think of doing a show have every friend of family member that comes in your house try picking up your cat and petting it. Is it friendly to everyone? Then it would probably tolerate a show and the up to 12 judges that will be handling it in a weekend. But if it struggles or hisses or acts afraid - no - absolutely NOT a cat you should take to a cat show.
TICA's website has lots of info about showing HHP's. One of my cats has been Best in the Northwest twice and also twice top 10 Internationally. She's on this page twice http://www.tica.org/public/hhp.php
TICA's by far the best to compete in with Household Pets because titles are handed out just like the pedigreed cats as well as Regional and International Awards. They also have the most about of HHP's competing so competition can get pretty fierce! To find a show near you check their calendar http://www.tica.org/public/calendar.php If you do enter a show enter in the "Benching Request" area that you're new and to please place you near an experienced HHP Exhibitor. Makes it MUCH easier to know when you need to get to a ring, how to track your points, etc.