The nitrogen cycle is the process where harmful ammonia (fish waste and decomposing uneaten food) is consumed by beneficial bacteria which produces less harmful nitrite. Different bacteria then consumes the nitrite and produces much less harmful nitrate which you remove with your water changes.
Cycling the tank is the process you go through to grow sufficient beneficial bacteria in your filter media to consume all the harmful ammonia and nitrite your livestock produces and convert it to less harmful nitrate, which as said previously you can remove with water changes. When your system is able to do this you are said to be "cycled".
A couple of ways to tell if you are cycled. Your fish will stop dying for one. Another way is to test your water. If your tests are consistently showing 0 ammonia and nitrite and you are seeing nitrate steadily rising that is a good sign you are cycled. It typically takes a couple of months to cycle a tank and there are various methods of doing this. Even if you do nothing your tank will eventually cycle.
Being able to test your water is a good idea as your water parameters can give you an idea of what is going on in your tank and act as an early warning of issues. Test kits are available, API freshwater master test kit is a good one. It covers the basics (pH, ammonia, nitrite, nitrate) is good value (it will do 100s of tests), and is fairly easy to use. I would recommend a test kit, failing that fish stores will usually do water tests for you. Some will even do it for free. If you get a fish store to do it, make sure you get the test results. Dont just accept that "they are fine".
Can you explain more about the soap and what you did? Soap is the last thing you want in the tank.
Can you post a picture of your "mold"?
As i said it sounds very much like diatoms, which is commonly called brown algae and is a bacteria colony feeding off nutrients. They commonly occur in new tanks and normally go away as the tank establishes and the nutrients balance out. Over time green algae will outcompete the diatoms for nutrients and green algae is easier to deal with. Diatoms are harmless, just unsightly. Get an algae scraper and just periodically wipe it off in the short term.
You also asked about an algae eater. I would go for a snail or 2. Perhaps nerite snails as they cant reproduce in freshwater. They will still lay eggs though which can be a pain to remove. Nerite snails will eat diatoms too.