DS is now 7 months and self settles most of the time both for naps and nightime sleep. I let him sleep on me for his daytime naps until he was about 5 months and co-slept until then as I was still bf-ing at night and found it the only way I could get some sleep and function during the day.
AFAIK, you can't teach them too much until they're at least 3 months, as Octavia says their brain can't form associations until then. And at 10 days your supply is getting established, and will do so for a while yet.
However, I do understand how sleep deprivation can break you and you need to take care of yourself too. As I said, the only way I could get some sleep was by co-sleeping, I understand this is not for everyone but after falling asleep unintentionally near DS I thought it would actually be safer to choose to co-sleep and follow all the safety guidelines.
It really does get better. Things change and as long as you try and settle your LO in her cot/basket every now and then she will not want to fall asleep on you forever. While I agree that for your sanity you need to gently teach them to fall asleep on their own, 10 days it's too early IME.
Regarding the night feeds, DS has always been a snacker and just like your DD he would suck for a few minutes then fall asleep on the breast, and want to feed a bit later. I tried and tried to keep him awake for longer, changed his nappy and put him back to the breast, etc, but he just fell in a deep sleep again and there was no way I could make him feed for longer. I just accepted it and things gradually started to improve, and feeds spaced out.
Sounds like you're doing brilliantly, it WILL get better.