With regards to getting a Thai passport for your daughter, this can be done at a Thai passport office, I am assuming that you are named as the Father on the Child's Thai birth certificate, under normal circumstances you would have to be present at the passport office for the application, however, you can provide a parental consent letter to allow your wife to make the application in your absence, unfortunately the MFA website is under construction right now, so I cannot provide the link.

As you are in the US, then you will probably have to go to the nearest Thai embassy/ consulate with proof of ID etc. and sign the parental consent form in the presence of a consular officer.

Without this consent, your wife cannot apply.

Alternatively, she could go to Thai immigration and ask if it is possible for your daughter to depart Thailand using her US passport.

Not recently,  but my four children all got passports in Bangkok without me being present, just their mother and birth certificates.