Dear Sir!
I worked with PHP > 9 years, done so many projects, include using javascript, jquery, api (Soap, RestFull...).
I summary your issues
- check all requirement of form to make sure they entered when form submit
- if picked up at airport, should have textbox to enter flight
- option to add car seat or car meet(car type)
- option choose greet services
- option to select type of vehicle (car type?)
- ajax load, check if new reservations comes, at back-end, play sound or notify, alert for dispatcher
- send email to customer email, with info you gave
- try to use API of sendhub to send sms to customer.
If you want site works stable, fix & done 1 times for all issue, you can trust me, i can do it for you.
Thanks!