You had to bring out the dirty word.... Chemist....
There's two ways you can approach fish keeping. The chemist way, or the K.I.S.S. method. Personally I subscribe to the K.I.S.S. method.
I never test my water. Wasted a lot of money on a master test kit and tested my larger tanks once to find out they were fine.
Your tank was set up before, that's a plus. You probably have plenty of beneficial bacteria in the tank, it's close to impossible to kill it all unless everything was bleached. Even rinsing the filter media under tap water will not kill it all. If that were the case I'd have a fish room full of dead fish. I rinse the snot out of my sponge filters when they look like they need it... I do it right under the tap. Yes, I have city water. As long as you do not use extemely cold or extremely hot water to do so you'll be fine.
The fact that you're keeping one of the easiest cichlids to breed and keep and the fact that they have already spawned leads me to believe your water is just fine.
Do 1/3 water changes from here on out for the existence of the tank and you'll be fine.
Now this doesn't mean to stop reading and learning... I've been breeding cichlids for 25 years now and I'm still learning something new all the time! I just don't want a new hobbiest to get scared off by being overwhelmed with testing water and such.
K.I.S.S. = Keep It Simple Stupid
Rant over

Enjoy your cichlids and enjoy this site, there's a wealth of knowledge here!