To me it sounds like he's contrasting the healthy cat with some other cat. His current cat is not healthy and would lose a fight with a mouse if it actually managed to catch one. So he says that tomorrow he'll buy a healthy cat instead. That's how I understood the English sentence.
If anyone wants to buy another cat because their cat is unhealthy, they would say, "Tomorrow I'll buy another cat". "Куплю здорового кота" definitely refers to the size, not health. I'm not saying that the English translation with "healthy" is wrong, but it is very unlikely and the version with "big" instead of "healthy" should be accepted.
Because nearly all neuter nouns denote inanimate objects their accusative forms are identical with their nominative ones. I can think of only two exceptions: чудище (a fantastic creature) and чудовище (a monster). Both seem to be animate; however, they act as such only in their plural form. Thus, we say, «Он отпугнул чудищ/чудовищ» (“He scared the weird creatures /monsters away”), but «Я вижу чудище/чудовище».
What determines whether you can use "going to" instead of "will"? This time I translated it as: "I am going to..." instead of "I will...", because there were similar examples in the previous lesson, and it sounded more natural. Should I have reported it as a mistake, or is there a particular reason it's stretching the translation here?
If we exclude from consideration verbs ending in -нуть, prefixless perfective words are rare in Russian. They include купить(ся), решить(ся), дать(ся), деть(ся), стать(ся), сесть, лечь and пасть. There are also verbs ранить and женить(ся), which can be perfective or imperfective, depending on the context
Yes - according to the previous comments that is an alternative - and, in this case, more likely - translation. People would not normally talk about buying a healthy cat (why would anyone knowingly choose otherwise?), but they might say that they were going to buy a big cat (without necessarily meaning a lion or tiger).