Green water algae is not the result of poor cleaning and water change practices. It is the result of too much light getting into the tank. Any tank contains enough nutrients for green water to grow, so if you supply enough light any green water that is present will grow and multiply quickly. Do not worry about the green water except that it makes it hard to see the fish. It will not harm the fish.
A bacterial bloom is not green so that is less likely to be happening.
If you have retained the thin sponge that is present in most cartridge type filters, you may still have a partially functional biofilter although it will not be as well developed as the bacterial population in the cartridge would make it. As others have already said, you may have set yourself back by discarding the old cartridge. It is worth at least testing to find out how you are doing. You need to test for ammonia and nitrites and reduce both to less than 0.25 ppm using large water changes. If you find both are at zero, it is possible, you have gotten lucky and have a functional biofilter in your thin filter sponge. Some of the newer cartridge type filters use rough surfaced plastic pieces instead of that thin sponge. My new Whisper filter has a plastic plate with lots of irregularities on it that the manufacturer says are there to act as a biofilter. The plastic piece looks very small to me compared to what I would expect to have as an effective biofilter but they are free to claim anything that they wish.
Cleaning the filter housing in the bathroom sink is a good idea as long as you set aside any filter media first. That includes the filter cartridge, the thin sponge if your filter has one and the plastic "biofilter plate" if your filter has one. The pumping device, looks like a tiny propeller, and the well it sits in should be scrubbed out well. I use a test tube brush for cleaning the well. Be careful handling that impeller for two reasons. It is very thin so it is easy to simply break and it has rubber bearings on each end of the shaft that are needed for the impeller to run smoothly in the filter. Those are tiny and can be lost quite easily unless you are careful.