I've noticed that the endings of nouns and adjectives, like здоровую and кошку in this example, end with similar sounds. I wasn't actually sure which word to choose in this answer so I went with здоровую to test out my theory and I got it right. Was just this just a lucky guess or is this an okay-ish way to figure out what declension the adjective needs to be in?
Does the Russian word здоровый have the same double meaning as "healthy" in English?
(1) having health (in oneself): a healthy cat
(2) bringing health (to others): healthy food
Greek has separate (but related) words; English, German, and Slovak have the same word - how is it in Russian?
It does. Здоровый may also be used "huge, hefty" (in informal settings).
Another meaning you forgot is "right, reasonable, natural" as in "healthy competition". It is not quite the same as "having health" but close enough, so both English and Russian use the word for this meaning.