Yes, we provided Mothers Birth Certificate and her two name change certificates, didn't see any reason not to provide it.
I'm not resident in Thailand so didn't provide anything with my name and address on.
With your point 11, where it says "a link to the parent applying", in our case, even though we are married, we don't live together, but Mother and Child do, so the parent applying was the Mother because they live together in Thailand. Only one parent applies. If you aren't married, but live together, I guess the Mother still needs to apply (and know someone who can countersignature)? If you are married and live at the same address as the child then as the father you can apply for them, but would need to provide something with same address as your child.
From the application form:
From the guidance document: