1. You must insert a language constant as the name of the new field and add this constant to the language constant override:
https://www.joomlashack.com/blog/joomla/languge-manager/2. I don't know exactly, but I think you have a badly set up administration.
An image for analysis would be useful - detail of setting the wrong field
But the reason will probably be different - a mistake in the template?
I would have to see the PHP code of the registration field template in the cart, which is called as a placeholder
3. with scripts? Scripts is piece of code aka Javascript. Really?
I think do you mean the same values as registration form, isn't it?
And what wouldn't the preview be?