Do I have to send my passport to apply for my daughter's UK passport and is my counter-signer problem due to our short acquaintance?

The process of applying for a UK passport for your daughter does not require you to send your passport; a copy will suffice. You will also need to provide your daughter's long-form birth certificate, proof of address in Thailand, and the mother's identification. Regarding the counter signer, it is generally acceptable even if you have known them for only a year, provided they hold a UK passport and can present a copy of the ID page.