Is this nervous laughter (a good sign?) or a polite way of saying no?

The guy I like...(CRAZY about really) lives next door, I am 30, he is older than me. He moved home to care for his ill mother. We hit things off because of our common bond of dogs, and I had some that he loves, as he can't have any right now now. So things seem awesome between us, we only walk when we are both outside, until about 2 weeks ago when I stopped by before my trip to wish him well on his surgery and then asked if it was OK to come by like that, he assured me it was and to come by any time, at any door... so I did again when I got home to see how he was feeling. Then yesterday we had a awesome day playing in the yard with my dog and I asked if it was OK I stopped by the day before and he again said, yes, come by anytime and he told me, his mom gets VERY excited when I come by and runs down the hall to tell him "that lady and her animal are here again" and follows him to the door. This is pretty amazing because he says she does not like people and never likes company over. I have NEVER seen anyone visit the house, even his best guy friend next door. Yet... when I asked him about a month ago to come to my dogs birthday party if he wanted, he kinda laughed and I asked what was so funny and he said "nothing, I just think its cool" and told him, to come if he wanted but no pressure he had games that day and never came. Then yesterday he tells me he needs to do more walking for therapy and I suggested him walking my dog, never even said I had to go with, which I don't, and he did the little laugh/chuckle again and said, OK thanks... but I don't know if he was serious or just trying to be nice and turn me down? But why doesn't he just say no then and why offer to let me come by anytime when no one else does? Why spend 30 minutes or more talking to me every time we see each other? I have read that nervous laughter can be a sign he likes me.. but I don't know. I can tell you a few other things he does often and maybe it will help... he ALWAYS gives me eye contact... this has been since day 1, he could just be that kind of person but I don't know many people like that. He "reanacts" things for me, like that happen at a game etc and I mean he will jump around, shake his butt, whatever to give me the "full effect" of the story, he laughs at my jokes and makes jokes around me, talks about his childhood and a lot of the issues he had with his mom, he tries to relate to me a lot by bringing up a lot of animal topics... I feel he might really like me but then, why is he too afraid to leave the yard and do anything with me? We have known each other 2 months now. He is also fantastic at remembering things I tell him and always tells me things he really wants me to know even though I remember them too. Like how he always tells me about his childhood dog who looks like my dog. So confused... maybe he just likes my dogs? Anyone?
Is this nervous laughter (a good sign?) or a polite way of saying no?
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