Looks like your spouse and children are eligible for a non immigrant visa based on your ED visa. Where are you applying for the ED visa? They will tell you exactly what you need to supply (marriage certs and birth certs) for the dependent visas.
why would they need to accompany you on your border run? Once they enter the country they could stay indefinitely. They cannot overstay because they are a minor.
I don't know what everyone's on about.. we didn't have any issues crossing into Laos to do a visa run in Savanakhet. Have your bluebook in your name and some photocopies, the car passport which you can get from DLT. No dramas.
Hmmm. I exited Thailand on passport A, entered Loas on passport B, then exited Loas on Passport B, then entered Thailand on passport A during my last visa run to Savanakhet.