Learn Russian: Сейчас мой кот худой, потому что он много гуляет. - Now my cat is thin, because he walks a lot.

This sentence uses the basic adjective худой (thin). Since we are learning comparisons, how would I say 'thinner than a dog' using the Genitive case?

First, change the adjective to its comparative form, which is худее (thinner). Then, instead of using a word for 'than', simply put the animal being compared against into the Genitive case. 'Dog' is собака, and its Genitive form is собаки. So, 'thinner than a dog' is худее собаки.

Does гуляет mean the cat is being walked on a leash?

Not usually! While гулять means 'to stroll' or 'to go for a walk' for humans, when talking about pets like cats, it normally just means 'to be outside' or 'to roam around outdoors.'

Why does худой end in -ой instead of the typical -ый or -ий masculine adjective endings?

This is entirely because of where the stress falls. Masculine adjectives ending in -ой are always stressed on that final ending. If the stress fell earlier in the word (like in старый - old), it would use the -ый or -ий ending.

Can I change the word order and say гуляет много instead of много гуляет?