Is it ok to pay a home daycare service under the table?

No. Not totally sure what you mean by "under the table". They have to declare taxes based on monies they take in, so it is illegal for them to just accept payment and not report it. There are forms you need to fill out that are filed with social services for a registered daycare. Now if you're just hiring grandma next door to watch junior for a few hours, fine. Anything more, make it "above board".
No. The idea is a bit confusing. If its a family coop and your clan has done it for years its probably done a lot& not as worrisome. But unlicensed facilities are that way for a reason. If they are working outside the law in this area what other shady things are they into. If something happened to your child at that facility you have little recource & may be on the hook for chld endangerment.