How does the notification of residence process work for TM 30 in Thailand, and will I receive a receipt?

You won't get any confirmation AND there is no print function for a receipt.

Have your land lord go into the system, use the SEARCH function access your data, do a screen grab of it then print that.

IN Bangkok it's ALL you need, you don't need to go get anything from the immigration office, they accept the print screen as your proof of having done a TM.30