Yes... same or next day per email... the printed version (original, stamped and signed) comes per snail mail to your branch... 3-7 days. Depends on you local Immigration if they accept the self printed statement or not.
Yes, any other Bank can do it "to go", just Bangkok Bank need to order it from the Headquarter in Bangkok (in the branch they can do only up to three month). Well... that's how I experienced it in the last 15 years, but maybe it changed recently... I will know it in about 3 weeks... :-)
Not via phone or email, that's correct. In the branch (normally same or next day per email or 2-5 days per snail mail) or in the app... "more services" -> "request account statement".
The immigration know that it is impossible to get the 12 month statement to "today" from Bangkok Bank, since it always comes from Bangkok per snail mail (up to 7 days old) to your branch... that's why they accept it. Or... you order it per email... than it's max. 2 days old. The other one, the "today" statement (Bank letter), has to be fresh from the day you apply, that's correct.
One week for the statement sounds like Bangkok Bank. If so... you do know that you can order the statement via the banking app? You'll get it within 24h per email. Just print it out then at the next copy shop.