For cycling, the API kit is just fine and what I normally recommend. No reason to spend more for a test kit that will get used for a few weeks unless you are going to set up a few tanks.
Just keep in mind that the ammonia test may read 0.25ppm even if actually ammonia is 0ppm and that the nitrate test will not be accurate if you have any nitrites.
Most likely they are either being processed at a rate faster than they are being generated or you have a nice population of Nitrospira bacteria that convert ammonia directly to nitrate. Using Dr Tims, either is possible.
Not a requirement. As long as you won't be adding sensitive inverts your nitrates are just fine. Nitrites are not important in a marine system.
I actually recommend adding chaeto (or other macro algae) prior to adding fish but after the tank is cycled. The biggest benefit is that it will process ammonia directly just like your now cycled rock. It may not do much to reduce nitrates initially since you may be PO4 limited. If you do have enough PO4 in your system this is very cost effective compared to doing a large water change for a system your size.
Just remember, no starfish, urchins, or anemone's until you get your nitrates stable. They don't necessarily need to be low, just stable.
Awesome thank you. I was thinking about it, and it really seemed like a good idea to get the Chaeto a jump start and reduce Nitrates, but wasn't sure it would actually work. Glad to hear. Sounds like I need to get some chaeto ordered.