, OP is not talking about how to stay. OP's daughter has to clear her overstaying by leaving and re-entering Thailand to be able to apply for a proper visa.
Usually medical extension must be applied before you go into overstaying. Children under 15yo is not fined for overstaying but get overstay stamp.
You cannot bounce at any northern Myanmar borders.
Depending on the border checkpoint, some borders just don't let you leave if you are overstaying for more than specific days. In this case, medical cert cannot do anything.
You might need to stay for 1-2 nights in Laos if you go Laos borders with Chiang Rai.
You better talk with services if you want to bounce at the land borders. (no recommendations from this group though)
That is totally correct stamp. Info in the VISA CLASS field says "Non-RE", which means you entered on a Re-entry permit with a Non-Immigrant visa/extension, in your case with Non-ED based extension.
Or, ask your employer to prepare enough documents to apply for Non-B at the embassy/consulate outside Thailand (which embassy is according to the employer's direction).