Hello Bob!
It's my pleasure to be able to help a fellow Brit with their planning!
I have made some enquiries and am informed that the parking lot for Magic Kingdom opens 1 hour prior to the opening of the Park each day. I am not sure of the date of your reservation but would suggest that you check the calendar for Magic Kingdom
HERE for the date in question to establish what time the Park is due to open.
I would say that you would need to leave yourselves at least 45 minutes to get to Magic Kingdom from the parking lot as once you have parked your car, you will need to take a shuttle to the Transportation & Ticket Centre and from there will have to catch a monorail or ferry boat to the Magic Kingdom itself.
If the parking lot at Magic Kingdom is not going to be open in time, I would suggest driving to Disney's Contemporary Resort and explaining the situation. You should be allowed to park their for up to 3 hours which will allow you plenty of time to get to your reservation. Magic Kingdom is only a 5-10 minute walk or about 20 minutes on the monorail away.
Once your little princess has been transformed at the Royal Salon, you would need to return to the Contemporary Resort to relocate your car to the parking lot.
To answer your second question, you can absolutely use a gift card as payment for your package at the Bibbidi Bobbidi Boutique.
I hope that this makes sense and has given you a back up plan.
Please come back and ask me another question if I can be of any other help and have a fantastic holiday!