what you've heard is wrong. In and out of Chiang Mai immigration for over 5 years - tshirt, shorts and flipflops and Ive never once had a problem. There are some immigration offices that are apparently anal about dress - Chiang Mai isnt one of then
In terms of the visa, it is is canceled from the date advised by the school in the letter - source, I did it one hour ago. In my case, it cancels May 14, so I have to leave on or by May 14 or I'm on overstay.
The rest - as to is it required or not - my new employer said no, my school (also your school) says yes. My take - if in doubt, get it done anyway. With the letter you'll need your usual passport copies and it's counter 8 at Promenada.
It's the company itself that is asking me for the letter(s) for the application, in their words "recommended letter or certificate employee" and in a follow up email they said "the letter of recommendation would be someone noting that partnering with you in freelance has been a formidable affair or that alike" with a further note that "more the number of letters that matter."
It probably goes without saying that English is not their first language :)