Did you move into your boyfriend's place? The child (and Dad, subconsciously) might still view it as Dad's place.

The little boy sounds like he is being a little boy. As such he doesn't need discipline (if by that you mean punishment) so much as he does reminders on how to behave (in public and to others). The best discipline (as in controlling oneself) is self-discipline, and children learn that best by gentle reminders and observing others modeling good self-discipline. So his Dad should be modeling that for him by such things as "Thanks" said directly to you and to son and to the clerk at the store; and also "Let's make sure to thank [you] for driving us to the store" said to the son right before you stop the car.

Being a parent is a full-time, joyful burden. I loved it. Some people don't want that job. If you know you don't, it is time for you to start dis-engaging from this man.

If however, you plan on sticking with this man, it means sticking with his son, too. And it might be best if you and Dad move to a place that will have a specific place for son (and maybe a park nearby) or if you cannot move, arrange a space that is specific for son, even if it is just a corner of a room with a table and a big cloth where he can make a 'fort.'

His lying and sitting near you, wanting to touch you, etc. is probably a combination of anxiety and a desire to be friendly to you.

If you want the son to get used to you and Dad being a unit, then you cannot allow anyone to kick you out of your own bed. You and Dad need to be united on that.

It seems to me you have made several excuses for this little boy's father. I don't think Father deserves so much of your consideration; you have rights and needs and wants, too. Either you are partners in a relationship or you are something else. It sounds to me like Dad is taking advantage of you.
The time to work out the details of how you are going to proceed should be when Son is not there. But I wouldn't let Dad wiggle out of anything; he's had nine years to adjust to being a Father and that child needs the best Father he can have.

I see nothing wrong with you scheduling some other activities for yourself during part of Son's visit. Maybe you go and visit friends, or shop for yourself, and then come back together with son and father for dinner. Son needs some normalcy. You may need to be the one to encourage Dad to provide it instead of playing Peter Pan to his own child. I hope this helps.