But I'm still confused as to how to get Processing to receive the temperature data and continually update the .csv file.
For one thing, reading from the serial port is not something that should be done in draw(). You should have a serialEvent() function that does the serial I/O stuff.
Why the Arduino starts with printing a carriage return and line feed is a mystery. Then, it prints variable length values, separated by a comma and some spaces. At the end, NO carriage return and line feed are sent.
The Processing application then expects a fixed number of characters to arrive. It then expects that the input will contain a 'H', as the first character. Well, the Arduino never sends an H, so Processing never does anything. That seems pretty consistent with your problem description.
Perhaps you can now see what you need to do.