Things growing in my new tank

The white fuzz is most likely normal, although it typically comes with driftwood. Just watch it. and if you do a water change try to vac it up. Good probability that it will be gone pretty soon on it's own. The Black spot who knows, but I might caution that if you are running the tank light you will want to stop that. Ammonia is algea food, so adding light during a fishless cycle is not a good combo.

Now for the PH: your interpretation is not entirely accurate, but not entirely off either.
It is good to know your PH in the tank as well as from the tap. Ph is an important measurement because it relates to several other issues and parrameters.
Ph is something you shouldn't try to adjust except for extenuating circumstances, and something that can be difficult to manipulate and hard on the fish if it moves around a lot.
Those points are the same for newbies and experts alike. In almost every case it is better for the fish if you don't try to manipulate PH.

Now extenuating circumstances.
#1. Fishless cycling consumes KH very rapidly. KH drops, ph crashes, bacteria dies and the cycle stalls. This is very frustrating, not to mention time consuming. Therefore it is good to watch your ph during you fishless cycle. Since there are no fish in the tank it is also easy to adjust accordingly to prevent a crash. If your PH in the tank is 7.3 you are probably OK at the moment, if you have a KH test kit it will be worht your time to get a reading. Either way you can easily and quickly raise KH (and subsequently PH) with a little bit of baking soda. this will help the cycle and keep things going well.
#2. low KH tap water. IF your tap water or other source has a low KH level, it can lead to problems. KH is the buffer against rapid acidification. I personally feel a KH of 3+ is a good idea in any tank, with non-planted/ non co2 tanks, most folks would say it doesn't need to be that high, but to me stability is important, and low KH levels worry me.
There are a couple of good ways to raise KH that won't cause the rapid swings usually associated with PH fixing. Baking soda, and crushed coral (or some other variant of calcium carbonate) both of these will increase buffer, and remain fairly stable. an increase in KH always spells an increase in PH.

The last thing to remember/think about is that most people who try to fix PH are trying to lower it. when you attempt to lower ph is when you usually get into trouble. Essentially adding acidifiers, and or KH depleters (probably technically one in the same) is the requirement to lower ph. and as said When the buffer is low the tank can and usually will fluctuate and possibly crash.
Dave