You should be able to do it anywhere I believe. That's supposed to be the easy part. The difficult part is usually getting everything legalized before going to Thailand and then translated for the MFA.
You can go on the website and fill it out for her 3 days before she arrives. It will be linked to her passport, she doesn't need to do anything.
And no, she cannot fill out a manual form at immigration. She can go to one of the kiosks and fill it out after arrival, but it's the exact same as the website, just doing it on their tablet instead of her own device.
I've seen multiple reports of people with everything you have, and the immigration officers even call and talk to staff at the school, and still deny entry.
You told immigration you NEED to be in Thailand to study. That you NEED a visa so that you can stay in Thailand to study. And now your actions are telling them you don't really need to be in Thailand after all.