What would concern me most right now is not HIS behaviour, but his parents. I don't see anything "wrong" with him from what you have described, except overindulgent parenting. And this, you are already aware of. The 12 to 18m stage can be a tricky one, in that these kids are moving from a baby- baby to a toddler who is expected (and should be expected) to do so much for themselves. And if mom/dad aren't doing that, it makes our job so much more difficult.
A couple of pieces of my advice;
I would also stop actually picking him up for the "no reason" crying. Pat his back, reassure him verbally and so on. If it escalates, then give him whatever consequence you use for undesirable behaviour. Remind him that "We don't scream" or what have you. If he calms down by being in the recliner chair on his own for a few minutes, then that would be fine with me. He may be getting so overstimulated, that he just doesn't handle it well.
If he is getting so that he doesn't like people "in his space", I would respect that, within reason. He may have been okay with your son loving on him before, but he's not now. And not everyone will be, so now's the time when your little guy learns that not everyone likes that. I know I have had dc kids that would not be okay with another small one hugging and kissing on them. Some people (and kids are people too) have bigger "bubbles" of personal space than others.
I think you are doing the right things, but just need to take a deep breath and pull out some more good old fashioned patience. Over time, he will get better, if you keep enforcing what you expect from him.
I would come down on the parents about 2 issues. First, the wipes. I agree with Reggio. Ask them at home time today to bring wipes tomorrow. IN PERSON. If they show up tomorrow, send them home to go get them. Tell them you don't have any left. If they try to say, "Oh can't you just use yours?" just tell them no, that you have for 2 weeks now. Don't be angry, don't be sorry, just be calm. Second, I would remind them that for him to get good at eating solid food, he needs practice at home too. It is counter productive to have him on "baby" food at home and solid food at your house. Let them know that you have one set of hands, and although it's reasonable for you to HELP the kids eat, it's not reasonable for you to hand-feed them. Again, not angry or upset or sorry, just straightforward.