I doubt they will, tbh. There are something in the order of 400,000 foreigners living in Bangkok, if every one of them has to go to Chaengwattana every time they return to the country it's going to be complete chaos. It works in most up-country locations because there are comparatively few foreigners living there (and they don't travel that much). In Bangkok, which has 1 immigration office and so many foreigners, the system will just collapse under its own weight.
I think - think - you can print them but as I remember it's 5baht per page, so it's probably much cheaper to print them before. And yes you can pay in baht
There is no format that I am aware of. You are required to show 'proof of accommodation' so if the letter shows the address of the family, the date of your stay and clearly that you are invited and will stay the whole time, then i would think you're ok (presumably the family can write such a letter in Thai and English?)
You have two options as far as I can see - 1) Get the family to write you a fairly formal email/letter, print it and bring it along with you or 2) Make a cancelable reservation at a hotel through any number of hotel booking websites, print the confirmation and cancel the booking.
There is no way for immigration to cross check the address you put on your visa application/arrival card with where you actually stay (Especially if you stay in Bangkok)
agree. No restriction. Just have proof of accommodation, onward ticket and 20k baht cash (any currency), in the highly unlikely event that you're asked