She will go to the Immigration Office at IT Square Lak Si to extend her Visa Exemption.
But since she will leave Thailand on the 31st and be gone for a week she may not get the 45 day Visa Exemption currently going on in the first place, but the regular 30 days. As far as I know it is set to go back to the normal 30 days at the end of March. But extending the stay is at IT Square Lak Si either way.
She is required to have a ticket out of the country before her exemption expires when returning to Thailand. But she may or may not be asked for it.
If she will have problems returning is something I can't really answer.
if something happens, your insurance might not cover anything unless you have a valid International Drivers License. Even if you have a license back home.