that's a 30 day visa exemption. Very different to a visa on arrival. You can extend a visa exemption in thailand, once for 30 more days. A visa on arrival, is only 15 day, and can't be extended
Yes you can extend your 60 day visa once for another 30 days. You'll probably need a copy of your TM30 (ask your hotel), Photos, and copy of passport plus 1900THB.
The TM7 (application form) is available at immigration. Each office has it's own versions, easier to use what they want locally, than do it all again.
You can request the letters, statements via your local branch. However it will take a couple of days and the letter/statements are only valid for 7 days from the date they were issued. So you have to get your timing right. You'll also need your bank book updating on the day you go to immigration.
The hotel in Bangkok will or won't file one whether you like it or not. Whether you need to do one when you return north is down to your local immigration office.