yes it's just which ever is more convenient and can deal with it quickly for you. You tend to find all the hospitals will have the ability to sort out any test if required quickly
you will needto go to local hospital and pay them to give you the required certificate. On the evisa website there is a form that needs to be signed and stamped by GP saying you dont have any of the diseases mentioned
Do your children have thai passports as well as uk passports? If so they can enter on their thai passport so would be classed as thai and not need a TDAC, otherwise all 3 of you will need to complete the TDAC