I hope is well and all goes well.
Depends on how small baby is and if baby growing properly. More details would help for better responses. 37 weeks is full term if doctor wants to deliver for safety of baby that's all that matters.
It depends why she is small. If they suspect that low blood flow from the placenta is preventing her from growing then getting her out is the best for her health. My son was in the small side and they kept watching him. Eventually at 35 weeks he didn't gain much at all from the last time they checked and his amniotic fluid was low because basically my placenta had started to gradually stop working. It wasn't able to feed him as well and then stopped replenishing the fluid. :(
Yes! If your baby has inter-uterine growth restriction it means that your baby is not growing anymore inside you and needs to come out.
That's sounds weird. Sounds to me like ur Dr, probably wants to go on vacation and ur due would ruin ur drs plans. I'm just speculating, cuz I've never heard that before. Howuch does Ur baby weight so far?