Hi there
I'll try to answer your questions but there is not clear cut answer really, it is all based on your own decisions.
However as soon as the dummy falls out he wakes up and we start all over again.
This is why Tracy suggested in her books that the dummy would only be used during the wind down and not throughout sleep so that LO can use up their reflex to suck hard then you remove it when the sucking lessens and they learn to fall to sleep without it. This avoids the problem of it dropping out and avoids the many replugs during naps/nights when LO can't fall to sleep without the paci (as a prop for sleep).
I'm now wondering if I should get rid of the dummy
SIDS guidance is that there is some evidence that it is safer for baby to have a dummy. Also that babies who are reliant on the dummy for sleep should not have it removed cold turkey until 6 months and the higher risk age has passed.
I would read up as much as you can about dummy use and make a decision based on info as it is a personal thing in my opinion. My own LO refused a dummy so I did't have this decision to make.
when we try to transfer him to Moses basket he wakes up after 10 mins or so,
This is covered in the BW books as a common problem. You will need to shush/pat for another 20 mins after LO is put into the moses even if he appeared asleep when you put down. This is part of the sleep training method to pat until deep sleep and you see LO melt into the mattress and the breathing change.
However at night after dream feed he sleeps very well in Moses basket.
LOs are often different at night so although both night and day effect one another it's not uncommon for LOs to accept the moses/cot at night when they fuss more in the day.
and start trying to get him to sleep in the Moses basket rather than transfer him?
It would likely be beneficial to get him down into the moses before he nods off. I found with mine he needed to be in the position he would fall asleep and wake up in before he started the seven mile stare. if I changed his position he couldn't find his object to stare at (we did have this problem and had to go through a new wind down routine with a different focal point). If you want to go ahead making steps towards independent sleep then I'd get him into the moses sooner. He is not too young you can do this from birth, you keep picking up when he cries and settle fully in arms (calm, relaxed, not asleep) and you never leave him to cry or to cry hard without being picked up. I started with my DS pretty much from day 1 although I didn't discover BW until he was 4 wks. He was sleeping independently at around 8 wks ish.
I should just do whatever it takes until he is 12 weeks?
It's really your choice, it as much (if not more so) about when *you* are ready for sleep training as LO. If you feel ready go for it

if white noise would help? Can that be a prop like the dummy?
White noise is not considered a prop because you don't have to keep going back in, you can just put it on and leave it. If it helps I'd use it. There are all sorts of colourful noises available (look on line, lots for free many at a small cost too for download and you can set a CD player to repeat) including white, pink, black...any you or your LO like is fine.
hope this helps
