First on the ammonia and cycling. You need a source of ammonia to start the cycle; many use some fish flake food, just a bit, added every day. The ammonia in the tap water may work here, I'm not sure as I have never had this and I always cycle with live plants.
The Jungle Start Zyme I am not familiar with; their website says it contains bacteria to jump-start the cycling, and they also say it breaks down sludge. The latter process will produce ammonia. I am not one who recommends messing with the natural bacteria in aquaria. Once the tank is established, the bacteria needed to break down organics will be there. I have no issue with pure bacterial supplements, though I cannot say if this one is true to its claim. I do know that Seachem's Stability and Tetra's Safe Start are both 100% live bacteria, and they do work to help start (not replace) the nitrifying cycle.
As for the ammonia in the tap water, with each water change you will need a conditioner that detoxifies ammonia. Some do, but not all; they will say this on the label. This will handle the initial influx of ammonia, and by the time the conditioner has exhaused itself (around 24 hours it will cease to be effective) the bacteria will have multiplied to handle things.
Your pH will work for many fish, but I still would like to know the hardness as this will tell us if the pH is likely to lower over time. The hardness is important for fish too, so we do need to find this out. Your water supply people will have this; no point wasting money on a hardness test that you will only use once.