Can I obtain a residency certificate with a visa exempt stamp in Thailand?

Sharing my personal experience at Chiang Mai immigration just last week.. my TM30 registration was same day no problem.. but this process simply acknowledges your residence in a private home.. it is not the 'certificate of residence' needed to open a bank account.. which you have to apply for separately on the 2nd floor of the main building.. and Bangkok bank made it very clear to me that i couldn't open an account without the pukka certificate.. the immigeation clerk told me that it would arrive in the post by 20 April.. and when i queried why that was 27-days away i was informed that that's how long it takes.. and there was no 'express' service available to me.. eg. upon payment of a higher fee.. so i simply have to grin and bear it.. no doubt others have had different experiences with different outcomes.. it seems to me that it's all a bit of a lottery.. in the words of forrest gump.. you never know what you're gonna get!

From your embassy, yes, depending on the embassy. Each one has their own rules. From immigration - depends. You greatly increase your chances if you're willing to pay for it and explain the purpose (for example, to open a bank account, apply for a driver's license or purchase a vehicle). Each immigration office has their own rules, but I'd imagine they'd be willing to issue one if you explain you need it to open a bank account, which in turn is needed to apply for an extension of stay (for marriage, retirement etc.)

Yes

It depends on where you live. Each office has different policies.

For example in Chiang Mai you can get a residence certificate on any stay, only need a TM30 and 500 baht to pick it up next day.

In Bangkok you can ONLY get a residence certificate after you've filed your first 90 day report, which you cannot do as a tourist and therefore cannot get a residence certificate in Bangkok as a tourist.