Hi,
It sounds like you are in the place that I was when my LO was really small. Keep in mind that you LO is only 3.5 weeks old - she will need help to learn to fall asleep for quite some time. Please, whatever you do, don't try the Controlled crying etc... esp. with a 3.5 week old - she's just trying to learn if she can trust you: if she cries, or needs help falling asleep, help her, otherwise you will have problems forever because she won't trust that you will actually be there to help when she needs it. That being said, I have a couple of suggestions:
1) Don't worry too much about schedules. At 3.5 weeks your little one will probably still be eating eratically, and sleeping at odd times. Feed her when she's hungry and don't try to force more food into if she's not, or you'll end up with bad gas pains to deal with (the voice of experience talking here).
2) Chances are your LO is falling asleep at the breast because that's what all newborns do - it's warm, it's comfortable and they've just had a shot of Oxytocin (acts like a sleeping pill for baby).
3) I know all about the not wanting to be put down thing - it took three months before my LO got over it. Some suggestions: warm the bed up first with a heating pad or blanket (but make sure you take it out before the baby goes in), and made sure the baby is warm enough once in there (enough blankets etc...). You can also try wrapping her in something that you've worn (and not washed). That way, the smell of you will still be there. Right now, your LO is likely still a little freaked out about the brand new world and just wants the comfort of knowing you are there. Don't worry, it won't last forever. Think of it as a chance to cuddle and relax - something you should be doing lots of right now anyway. :-)
Let me know if any of that helped, or if things are getting better.
Theressa