I think the weirdest thing for me is him suddenly wanting to cut the grass, in the rain, when previously he has put it off because of the rain. I don't actually know anyone who cuts their grass in the rain come to think of it.
Add in the presence of the neighbour when he was standing there crying. OP you also said neighbour didn't seem concerned he was crying. Most people surely when seeing their male neighbour in tears in the rain, would look concerned if they approached and talked to them.
Add in the info that said neighbour was friendly and helpful when you first moved in, but has since essentially backed off anything above a simple 'hello' to op, but converses with her DH with no issues.
Add in the now gaslighting, denial he was crying, anger at being asked about it, anger and blaming OP for bringing it up to the point of not speaking and sleeping on the sofa.
Theres also the fact this man appears to do nothing in the home (save grass cutting), nothing with the DC, does not sound like he cooks dinner and has multiple hobbies where he is frequently out of the house.
He also refuses to take phone calls from OP, but has no issues taking calls from other people.
I am a strong beleiver that nothing is a coincidence. Why cut the grass in the rain, and when neighbour is outside, and be crying. There is a link here between neighbour and OP's DH. Whatever it might be, but the most likely is they are or were involved in some way - whether a relationship or involved together in something else entirely (financial/drugs etc - how likely are these other things OP?)
Based on that my guess is he has a nice comfy life with OP, she does everything for him, he gets his clothes washed, his food cooked, his life sorted and has no intention to leave. But has been seeing the neighbour for some time, recently they have had problems and she has ended the relationship. He begged to talk, they met in the rain under the guiase of him cutting the grass, and he is begging her, hence why he was showing emotion and crying. And why she seemed unaffacted by it. Or she is fed up of him refusing to leave OP and has issued an ultimatum and he was panic crying.
None if it looks good though, whatever it is.
Even if its simply, he was crying over something totally nothing to do with neighbour, he is still lying about it and being angry at OP for daring to mention it.
OP, given his reluctance to actually speak to you, and denial of the crying, and presence of the neighbour I would be checking his phone at this point. her is hiding something, not telling you, if you are supporting and housing this man you have a right to know what it going on.