Chiang Mai Immigration has returned to near the airport - if you know where the airport is immigration is not only well signposted it's about 250m before you turn left (same side) as the airport turnoff.
TM7 form, copies of your passport details page, entry stamp and departure card, a 4x6 photo (usually with blue background) and 1,900 baht payment. Any immigration office.
Timing depends on the office but most will allow you to apply at any time - your 30 day extension starts from the day you were meant to leave Thailand not the day you apply.
I get that flights to Yangon are dead cheap from CNX at the moment (1490 was the price I saw recently) but you're not considering that most who go to Vientiane fly to Udon Thani and bus the difference. Also you can get a VOA for Laos, you need to apply for a Myanmar visa in advance.
Cheapest destination direct without requiring a visa would be Kuala Lumpur. There are cheaper flights to Yangoon but a visa is required to even get on the plane.