I'd be willing to bet that your CYA is, as you assume, a large contributor to the problem. But, in order for you to know for certain, you need to test your water with a reliable drop-based (no test strips please) test kit. Consider the TF-100 available at
www.tftestkits.com or a Taylor K-2006.
If you have been using stabilized chlorination products exclusively for a period of time, I have little doubt that your water is overstabilized. Stop using them now, test your water, post your results on this thread, and the experts here will help you get your water clear. While you are waiting for your test kit to arrive, take a water sample to a pool store and have them run a set of numbers for you. That will get you started.
When using bleach, you are correct in assuming that you simply add it directly to the pool. Some pour it in a skimmer and this works great unless you have a skimmer sock over your basket. I learned that skimmer socks turn to mush after a couple of rounds of straight bleach. You can also pour it slowly in front of a return jet.
As for lowering CYA, the only way that one can willfully lower it is by removing water (draining or splash-out).
No worries. Fixing a green pool is pretty easy once you understand why they go green in the first place. What's better is that with BBB, you don't have to spend $$$$ at a pool store.