Do I need a new COE if I change my departure city for a flight to Thailand while keeping my final destination the same?

Im pretty sure you can only transit through Singapore if you are travelling on one ticket, check this.

It’ll depend on what flights are listed on your COE! Is your internal flight listed? But most likely yes since the flight to sing would be listed on your COE.

The info on the COE changes therefore get a new one. I had quick turnaround for COE. Little errors on COE, like wrong date for arrival at Tokyo connecting point, were tolerated at transit checkin at Tokyo but if important info was wrong they might not tolerate it. I would not risk it. I would obsessively check accuracy and acceptability of documents and assemble them into a bundle with an index on the front. I saw several people in Tokyo delayed sorting through a messy bundle of documents.

100% yes

Yes I would

As Survanabhumi and ASQ asked me for both my boarding passes from Sydney to BKK stopping at Singapore

I think so.

I think your COE lists all your international flights, from exit to BKK. In case contact tracing is needed.