I think first of all, if you want your sister and mother to quit bad mouthing your boyfriend to you, you have to quit supplying them with ammuniton. I mean you don't have to lie to them but you don't have disclose every negetive thing going on. I understand you probably need to vent... you should do that in a journal, or here or to a non-judgemental friend, but keep mom and sis on 'need to know' basis with whats going on in your relationship.
If you love this man and want to be with him ... you have to understand that he's not likely to change much. I mean he may get a job eventually, he may (hopefully) get to a doctor and start getting treatment for his social anxiety... but who he is, his pace of living, the laundry thing etc... the way he is in general (sort of lazy) is part of his character and will probably remain so.
One thing I got from everything you said was that he seems to care about you a whole lot. He quit smoking so that you'd date him. If you asked him to do something for you, lazy or not, he seems to be able to get it done for you (the peanut butter sandwhich example). It seems like if its for you, he will get off his rear... but doesn't have that motivation to things for himself.
I wouldn't be surpised if his social anxiety is what is preventing him from seeking employment, from dealing with the unemployment people, etc. That seems like something that should be addressed before everything else as it may greatly improve his quality of life.
Can you help call around and find out what kind of low-income medical assitance plans are available at some of your local hospitals? A lot of them charge on a sliding scale and if he has no money... hey may be able to qualify for some sort of low-income health plan with a very minimal co-pay.
Getting in to a doctor for a complete physical and to discuss his anxiety may help get him on to a better track. Like I said... this isn't going to change him, or make him suddenly want to fold clothes and clean his room... but it may help him to become more functional for work and such if he is getting the kind of treatment that he needs.
I understand your frustration, and also your love for him despite it. It sounds like you are driven and have many goals and it sounds like he's the opposite of that. Perhaps he will become inspired by you to get moving in the right directions, hopefully he will. But if not you will probably have to face the fact that this is the guy he is, and you will have to love him for all the things he is and isn't or decide that you can't.