If I have a landing page that ranks for Cat Food, and I have a subpage off of that page that is optimized for Cat Clothing and I get inbound 'cat clothing' links pointing to the cat clothing subpage, will it help my main page rank for Cat Food?
An inbound link to your cat clothing subpage will boost that ranking of that page. And if you have links from this page pointing towards your Cat Food landing page it will boost the ranking of your Cat food page as well. More over if all your incoming links are towards the index page and inbound deeplinks it looks artificial. Having some deeplinks to your inner pages will look more natural and it will benefit the overall site ranking.
if both cat food and cat clothing are linked (which I am sure they are) it should help you for sure (very little though). This whole link and ranking thing is way too complicated, just make sure you get your priorities right.