You don't get awarded a kill in the following scenarios:
- If you knock someone down but they're still crawling when you yourself are knocked out of the match
- If you knock someone down, but their teammate revives them before they're knocked out
- If someone else knocked an enemy down, and while they're crawling you're the one that knocks them out completely. In this case, the kill goes to the person who knocked them down and you get an assist.
Also, separate from my original question, I'm pretty confident that damage done to someone already knocked down (regardless of if you're damaging their knockdown shield or their 'knock down hp') doesn't count towards your match totals. I don't have confirmation on this either though.