For Thailand, always print everything. The email you received with your visa should have instructed you to print it.
Hand it over with your passport and final boarding pass when you get to passport control, then when you step away, check the stamp you received to make sure it's correct. Do this every time you step away from passport control.
And don't forget your TDAC when you're 2-3 days away from traveling.
It's an extension. It extends your time. It's added on to the end because the extension you'll be applying for will be for the reason as the entry stamp you receive.
the only time it needs to show as international transfer is when you're trying to convert from tourist to non-O visa based on retirement inside of Thailand. The one year extension doesn't have that requirement.
You cannot get a DTV inside of Thailand. It's only available from Thai embassies through the e-visa system.
It's also not as simple as just applying. You need to qualify first, and then provide all the paperwork as part of your application showing that you qualify.
Make sure you get the proper paperwork from your employee stating the end of your contract and last day of work, because you will need to go to immigration yourself with your passport and cancel your work extension to properly close out the loop. If you don't do that, you could be limited on visa options in the future and fined up to 20,000 baht for failing to notify immigration that you stopped working. Even when your extension ends, you are still required to report the change of status to immigration and officially cancel your extension with them. You can do this a few days early and they'll still let you stay until the date in your paperwork if it is on or before your extension ending.