Schedule and consistency is obviously an issue. He really doesn't know what to expect when and has learned if he fights it that he wins - parents admitted to this.

That doesn't mean he gets to win at daycare. I would leave him in the playpen but make sure it is a very safe place should he decide to climb out. No toys in bed it is sleep time not play time. I don't even allow a stuffy in bed cause it is dangerous and most have grown up without one as infants so no reason to start it as they get older thinking it will keep them sleeping - it keeps them playing.

Yes they almost all go through a stage around 18m where they flip the mattress - usually by then the velcro tabs that hold the mattress down are useless so it is easy to do. Just put the mattress back in place and lay child down, give them a light sheet/blanket either saying nothing as you do it or just repeating it is naptime, lay down and go to sleep. No other interaction. Will waste 1-2 weeks of your quiet time but after that it usually begins to settle down.

On the arrival time there is no reason for child not to arrive at the time you want so state what time he needs to be in care by. I do not allow a child to arrive after lunch because it doesn't give me the freedom I need to put cranky kids down for naps early. I want them to come so there is half hour of playtime then lunch then nap so they are in our normal routine. With their poor sense of time, a half hour of playtime is sufficient.

There may be no other option for a the off weeks but to let him sleep all morning and be up in the afternoon which is probably what mom is doing.