I understand your frustration with feeling like the only one cleaning. But at the same time, I've learned over the years that if I want my boyfriend to do something, it's probably not as obvious to him as it is to me. This means I have to gently point it out to him. But I"ve also learned that just because I tell him I want it done does not mean it will get done right then and there. If I really want him to do it, I have to let him do it on his schedule. Though, he does need some reminding from time to time, and even he'll admit that! But remember, if you want something done "right" (aka, your way), best to just do it yourself! That's a lesson I've learned the hard way myself (the number of times I've rewashed dishes ain't small )

A side note, my boyfriend and I never throw something out when it was the other's unless we know for an absolute, 100% fact that we can throw it out (and even then, we both feel more comfortable letting the other still throw it out). What if you had decided to return the shirt that the tag belonged to and he HAD thrown it out? You might have gotten upset then too, because it was yours and not his to throw out.

I would mention that you would like some help cleaning up around the house. See how he takes it, and if he starts to help out more. If he doesn't or you would like him to do more/something specific, then I would suggest talking to him, gently and patiently, and try coming up with an agreement on who does what and how often it gets done. Then write it down if you have to and both of you just try to stick with it.

Good luck!