Hi S.. Hope he is feeling 100% better now!
I am one of those moms who insisted that my children learn to put themselves to sleep. It's called self-soothing.
From your description of him pushing you away, it sounds like he actually wants to, but being a baby, still wants you around. You have to decide what part to play in his bedtime and naps. I opted for walking out the door and expecting my kids to fall asleep on their own, once I had met their needs.
We had a bedtime routine. It did not last long. No TV after dinner, period (it revs them up, and my kids didn't watch any TV at 10 months, just to let you know.) Bottle, brushing teeth, story, quiet music. Nightlight. Laying baby down, saying night-night, and walking out.
You don't have a crib, so he crawls over to the door. You cannot let him open the door. If he were in a crib, he wouldn't be able to do that. So you must let the room be his crib. Make sure it is very safe - nothing to hurt him. Then let him cry until he gets tired of it and goes to sleep.
If you cannot do this, then sit in the middle of his room with your back to him with the lights off. Do not talk to him or hold him. If he gets out of his bed, get up and put him back in his bed and sit back down in the middle of the room. He will get very bored and finally drop off to sleep. The first several nights, you might have to sit there for a long time and keep putting him back in the bed over and over. After a week or so, he should sleep quickly. It is very boring for mom to sit there like a stone and not talk or anything, so that is what you must be.
If you are taking care of your daughter, you might have trouble doing this during nap time. I would recommend leaving him to go to sleep. But make it so that he can't get the door open.
Dawn