As far as I know you still need to do it. Tourists generally don't need to worry about it unless you're going to transition to a Non-immigrant visa, then some offices ask for it.
Legally, yes the ministry of labor requires a bachelors degree to issue a work permit. Tons of people work illegally without one though, unfortunately.
Should you? That's entirely up to you. If your COE is already issued then you've already booked an ASQ in Bangkok. Is it refundable? Is your flight to Bangkok refundable?
Changing the COE is easy, just email the embassy. You'll probably need to apply for a new one.