Can I apply for a multiple-entry Non-Imm 'O' visa based on having a Thai child, and are Savannakhet and HCMC options for this?

I don't believe that Savannkhet or HCMC ever issued year-long, multi-entry Non-O visas based on being the foreign father of a thai child born out of wedlock. 😕

Unless I'm mistaken both places only issue single entry 90 day Non-O's.

I know people who would get the 90 day Non-O from Savannakhet then take a 60 day visit family extension and then go back to get another 90 day Non-O from Savannakhet, and they did that for years pre-covid.

That means your ONLY choice is to take a 90 day single and then either meet the requirements to get the year extension inside the country at your immigration office (400K baht in a thai bank account in your name only) or take the 60 day visit family extension and then go get a new 90 day Non-O

As far as can you get a work permit on that visa, Yes, if you find a job the employer can give you documentation to get a work permit from the MOL on that visa.