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.