What documents do I need to submit for my ME tourist visa extension in Thailand?

First of all, each extension application is a new one. You need to do whole things for each application. Whether you have a multiple or single entry visa doesn't affect your application for a tourism extension. And the Thai embassy in another country has nothing to do with the immigration business inside Thailand.

Q. do I need to fill out and submit the same papers as for a normal tourist visa extension?

-> You have to prepare all required documents at your local immigration office. no exception. Whether you have METV or SETV doesn't matter.

Q. do I need a letter from my landlord again?

-> If you mean TM30 receipt, yes, you need one.

This is general information for a tourism extension. You need to check your local immigration office for their own requirements.