Lets think for a second. As a note your alk should not ever go anywhere near zero. It needs to stay in the range 8 - 10, though often in newly mixed up saltwater it can be higher. Don't worry, it will drop with time (weeks, months) worry about it being low, not high.
I am not worried about nitrate unless it is very high, 50 or more. I am not especially worried about nitrite either as it's not actually toxic in salt water till it hits crazy levels (200 +, before that it's bound by a chlorine ion), but I don't like it that there's ammonia in the water. How much is it testing as , trace, or lots. Ammonia test kits are not especially accurate (and not nearly as accurate enough to say today .5 ,yesterday it was 1), so use it as a yes/no or no/little/lots test kit. I would bear in mind that when you mix fresh saltwater up it will often test positive for ammonia (a good reason not to use freshly mixed water).
If I was you, and had a tank, with saltwater standing there, with mixed up water and a mysterious ammonia reading I would take out 10 ml of that water and get a shop to retest it. I would also, assuming there are few bacteria in the tank, buy myself a piece of live rock in the one or two pounds range , and stick it in, using at as a source of many bacteria. If there really is ammonia , they will start to process it, if there isn't, no big deal. Then in a few days,, 3 or 4, get a couple of hermit crabs and see what happens.