I can’t help too much with your questions but I do know that it is possible to pick up/deliver embies yourself.
Ours are currently in sotrage at our RE office but we’re looking at transferring them to a long term storage place. We can rent a shipping tank, pick them up, drive them to the storage place and then return the tank.
This might be because they are already our embies, although I assume with EA when you are selected they become yours even before they leave the facility. (I know an adopted embie is yours I’m just not sure the point at which custody changes).
Also I know from shipping donor sperm from storage to the clinic, they only needed a days notice to ship it locally and overnighted it when it was going out of state. Again, I assume the same for embryo shipping.
Sorry I wasn’t more help. But good luck with everything !!