No access at all. The only time an airline would have some access is where they have a code-sharing or booking agreement in place with another airline.
It can never be emphasized enough - while a requirement that can be asked at any time, 99.9% of people are never asked. Those that are asked are mostly those who have come in and out of Thailand on a regular basis. The check is an excuse to deny entry to people they suspect of living and/or working in Thailand on tourists visas.
An extension is 1,900 baht. In Chiang Mai it's about 400-600 baht (roughly, sorry, been a few years for me) to get a VIP return ticket on Green Bus from the Arcade Bus Station in Chiang Mai to go to Mae Sai in Chiang Rai Province. You then get a Songtaew from the bus station (15 baht) to the border and cross into Myanmar and the town of Tachilek (500 baht or US$10 for the entry) then return and get another 30 days visa exempt entry.
Note that you can only get two visa exempt entries via a land border per year. Tachilek is a semi-nice third world shithole, you can literally be stamped into Myanmar and out if you want to without passing customs (their English is excellent) but Tachilek is quite the mini-adventure, some people highly rate the market, I don't :) The casino on the hill on the other hand is so weird it's worth a visit :)
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.