My babies have never slept on their backs. They just aren't comfortable that way. For the first few weeks, baby always slept on my chest. Then, I move them to right beside me (with my arm around them) on their side or sometimes rolling onto their backs. But when they sleep independently in their own bed, I have always put them on their tummy. Not what doctor's recommend, I know, so this is not something I am suggesting you do. It's a personal choice. You can try using a wedge that elevates baby's head when they are laying down. Might take some of the weight off the back of their head (gravity). Or, try putting baby down on his back with his head turned one way one night and then the other way the next. Personally, if baby's head is starting to become flat, my concern would be that baby isn't being held enough, but that's just me, please don't take offense.
As for tummy time, now is a perfect time to start. You could have started earlier, but it's not too late or anything. Baby will let you know when he's had enough. Some babies don't like tummy time at all at first. So, start with just a minute or two at a time. They'll start to get used to it and like it more. With my babies, I would put them on their bellies on top of me and then talk to them trying to get them to lift their heads to look at me. You can do the same thing by putting them on your bed and sitting on the floor so that their feet are away from you and their head is pointed at you. It's so fun to see them trying to turn their eyes up far enough to see you. With both of my babies (one is almost 2 and the other is 7 weeks, by the way), they would open their eyes really wide and wrinkle their little foreheads, lol.
Have fun exploring this new physical position with your little one!
God bless,
A.