What are the requirements and process for obtaining a residency certificate in Bangkok on a Non-O visa?

The requirements may be referred to immigration to immigration but in general they need those two documents you mentioned plus rent contract. I'm sure with this immigration will issue residence proof

Chaenf wattana &  you probably need a copy of your lease & maybe other documents 

I did this recently at the Samut Prakan IO, so it might be different. This was for car and bike licence renewal. I learnt a lesson here. 5 years ago the Samut Prakan office insisted I needed 2 certificates, and charged me 2x 500 baht. At that time, the DLT said one certificate and a copy were ok. Fast forward to now. DLT said one certificate is enough, but expected the wording on the certificate to state both car and bike licence renewal.

Again, Samut Prakan IO insisted I needed 2 certificates. I made the mistake of insisting I only needed it for the car. This time they didn't charge anything, but I had already decided to get one certificate (my second mistake). This office issues the certificate a week later. Went back to DLT and they needed the correct wording so I only completed my car licence. Went back to Samut Prakan immigration and requested the second certificate.

Lessons learnt here. This IO has improved its practices, although one certificate with both vehicle types stated would be optimum. Another lesson learnt is that DLT can only accept a single medical certificate if the licence applications are completed on the same day. Another original is required if doing the other licence on another day.

I hope this is useful for anybody reading. There were some bemused observers during this process, myself included.

I didn't think they were posting them out any more?

You go out to the main immigration office at the government complex Building B on Chaengwattana Road, you go up to the 3rd floor (the floor above the main immigration office) and there's the 90 day report/TM30/Resident Certificate room

You need copies of

Data page of your passport

your most recent entry stamp

your visa (the print out of the PDF visa if you got it outside the country)

your TM30

you will fill out the form,they will give you a small receipt and you come back the following week to pick it up.

EDITED TO REFLECT CORRECT INFO

You can get a certificate of residence in Bangkok on ANY stamp as long as you have a current TM30 filed showing you are registered as staying at an address in Bangkok