No. The -o ending shows that you are talking about yourself. For example if you just say "Bebo" people will know you're talking about you because of the -o. Eu falo Ele/Ela fala
Are verbs not conjugated differently for "I" if the speaker is female? Is that a rule across Portuguese as a whole?
Verbs dont carry gender, Whether you are male or female, you would say "Eu Falo..."
What about Obrigada/Obrigado? Or are those literally "Thanks" as in the noun as opposed to "I thank you"?
I thank you = eu te agradeço. (in a literal way).
Verbs in participle tense agree in gender and number with the noun/pronoun. (not in every single situation, though)
Obrigado/a is the short for "eu me sinto obrigado/a (partciple) a retribuir seu favor".
The word Obrigado/a is based on the person who speaks it. For example, If I am a woman then Obrigada is used (always). If I am a man then Obrigado is used (always). It's based off of your gender. Words learned from my Portugese friend.