People start queuing up about half an hour before it opens. Someone checks all your documents are correct (make sure you have a copy of everything required) and then they give you a form to fill in and a number. Then it's a matter of sitting in line and waiting for your number. Please note that it takes 3 working days to process, so you need to allow more time.
Visa extension for 1900 baht at immigration is probably best/cheapest option. 30 days. Can go now. Otherwise if you happen to be in Mae Sot or Mae Sai you can cross the border on a day pass for 500 baht.
Agree with this - even on a 6 mo METV you only get stamped in for 60days so you still need to extend for another 30 days at Immigration and then do a border bounce to get the next 60 days and then a further extension. Unless they will be staying right on the border and so can just bounce in and and out each time for a new 60 days, without the need to travel to a border or extend at immigration, it will probably end up being more expensive than 2 * single entry visas and 2 * 30 day extensions.
The other thing to note is that from most countries you can get stamped in visa exempt for 30 days - depending on your travel dates outside Thailand this may work - you simply get a second visa exempt entry on your return. Please note that airlines in some countries may want to see you have a return or at least onward ticket to allow you to board the plane to Thailand).
Yes, make sure you ask for it at the airlines counter when you're checking in or you will have to go back there again to get one. They don't have them at the immigration area.
Tip for the future - ask the first. Hotel you stay in to staple it into your passport for you.