I'm a massive advocate for co-sleeping - there are so many benefits I struggle to believe it is anything other than exactly what we are meant to be doing.
On your chest isn't great as others are saying but I also don't think it's as terribly dangerous as being made out. Your skin to skin contact helps regulate baby's temperature and heartbeat, and when you're that close IME you tend to find you have an awareness of your baby even when you are asleep.
It would be better to be side lying, you can still be tummy to tummy. At first I used to have to roll over when I wanted to swap boobs, but in time I developed a bit of a technique of rolling closer or further away to access the top boob. This way you can doze a bit and still feed and have that cuddle time. The images shared are good.
I was told 12 years ago with DS1 that side sleeping for babies in this position is safe, and the only reason they don't recommend side sleeping is that it's very common babies will roll onto their front from the side, and so people try to mitigate this by propping the baby up with things like blankets or muslins, which are a suffocation hazard. I can believe this, but of course midwives don't always necessarily speak the whole truth, and it may be out of date.
Agree, no swaddling if you're bedsharing, but it's OK if they are firmly in their own sleep space.
I don't believe you can "teach" a baby to self soothe by leaving them to cry any more than you can teach somebody to swim by throwing them in the deep end with no instruction. There are gentler ways. Establishing sleep as a cosy, nice, welcome and unthreatening thing is a lovely way to start teaching a baby to (eventually) self soothe.
Good luck!