The only correct sentence is the 4th one, and even that one is rather awkward. The 2nd is possible but strange, and the others are just wrong.
There is no need to try to use a passive form here - you're trying to combine a causative with a passive and ending up with something that doesn't work at all.
Here are some possibilities:
1. 'John let his son clean his car.'
This means that the son wanted to clean the car, and John allowed him to do this.
2. 'John made his son clean his car.'
This means that the son didn't want to clean the car, but John forced him to do it.
3.'John got his son to clean his car.'
This has a similar meaning to number 2, but without the element of force. This is actually the commonest one used by native speakers in everyday 'real' situations ( i.e. outside of grammar books).
4. 'John had his car cleaned by his son.'
This is the only natural way to use a past participle in this context.