For your first question, yes of course! Using the mother’s surname is the norm if the parents are not married. In this case, your baby will use your surname and will have no middle name - this will be blank. Also, the name of the father is usually left blank on the baby’s birth certificate.
For your next questions, you can authorize any person to file your baby’s birth certificate in your behalf in the LCR where you are residing. The place of birth will be indicated in the birth certificate itself.
For your reference, you can refer to these links:
REGISTRATION OF BIRTH CERTIFICATE: https://psa.gov.ph/civilregistration/birth-certificate
PERTINENT PROVISIONS OF THE LAW ON LEGITIMACY: https://psa.gov.ph/civilregistration/legitimation
I hope my answers and these links will be able to help you. 💋