The steriods will help A TON!! I would be surprised if the baby lungs WERE'NT mature, since she has had steroids for a while (sometimes they give them just hours before birth and it makes a difference!)
Is she in a major hospital with a NICU? I hope so, and it sounds like her doctors are doing what is standard. 37 weeks and after is considered FULL TERM, so even though technically she is 5 weeks early, in developmental considerations, she is only a bit before "full term". Of course, they will want to do that amnio, and they should. This will give them the best info of whether it is safer for the baby to be IN or OUT.
So you know, they will have a Neonatologist (NICU) doctor and nurses on hand for the delivery, and the baby will probably be "whisked away" for just a bit before your daughter sees her (maybe just in the corner of the room--- if she is doing well, to the NICU if she needs a bit more help). It is nice to have some idea what may happen, so I thought I would let you know!
We have spent a ton of time in hospitals (my son had his first open heart surgery at 4 weeks old and stayed in PICU with tons *tons* tons of complications for his FIRST YEAR-- so we know how that all goes....ugh) and know that doctors and nurses sometimes get a bad rap, but the vast majority do all they can to make sure all goes well--- so trust them and say your prayers! And thank God for all the medical advancements that are made every year..... I know we do!
Take care!