Haven't used KuCoin or COSS yet, but I definitely plan on using KuCoin in the future (when it happens to be the only exchange I can get a certain coin on, I'll sign up). Overall from experience, I've heard lots of positive things about KuCoin, whereas over negative things about COSS. I'd put my trust in KuCoin.
Also, there is a new type of exchange that will be opening soon called "Next.Exchange". I suggest looking into it because there's already been a lot of talk and hype surrounding the grand opening.
Is next.exchange part of NXT?
No I don't believe so (or at least never saw anything suggesting/advertising it), but I'm fairly certain NXT will be available on Next.Exchange when it opens or in the future. Nonetheless, I have high hopes this new exchange will be a great alternative to the current options out there.