Meaning of "that's fine" in modern American English when used as an answer to a question

The meaning of "That's fine" can be different in different cultures. Here in Australia, in the context you've described, it doesn't have to have a negative tone. I often hear it used as an iteration of "Alright". So, really meaning "Alright. Yes, please." or "Alright. Go ahead and send me the receipt."

Here in the States it is tricky due to the melting pot of cultures so "thats fine" could be interpreted as a simple acknowledgement/agreement to a passive aggressive way of showing ones upset/half-hearted agreement without actually saying the words. It all depends on the conversation. I would say in a professional setting, in general, it is used as an agreement/acknowledgement. Used in a personal setting its more looked at as rude... "can you watch my daughter for an hour, id appreciate it?" "Thats fine." This is a half-hearted or resistant agreement which comes off as "I really don't want to but I will so I don't appear selfish or rude."