Questions about example sentences with, and the definition and usage of "Bates"

A: "I can't call her Bates." means I cannot use that name for her.

"I can't very well call her Bates." means that I could use the name "Bates", which would solve the problem of what to call her, but it would create some other kind of problem. Perhaps there is somebody else who we already call by that name. Perhaps Bates is an inappropriate name for a woman. Perhaps Bates is a name that is insulting in some way. Perhaps she lives with a man called Bates but they are not married.

"I can't very well call her Bates." means that if I call her Bates, everything will not be "very well".

A: Survey is a type of research.
It asked a question along the lines of, "do you think there is discrimination in the hiring process at Harvard? To different people...

The reason he used a survey is to find what everyone feels about it, not just specific individuals.

A: Either because it sounds old-fashioned or because it's a complex word. So either a 17 year old is too young or too uneducated to use it. (I'd guess too young, personally, but I haven't seen Bates Motel.)

A: You could actually substitute "forever" as well and the sentence would still work. In fact it would be even more of an unusual and poetic choice of words than "ever".