Is line training (pay to fly) allowed in the USA?
It is not prohibited in the US, but I don't think it's currently practiced anywhere.  There used to be a school called Tab Express that did something like this, but the most notorious one was an airline called Gulfstream International Airlines that operated as United Express in Florida.  Gulfstream is now operating as Silver Airways and, as far as I know, has discontinued the practice of having the FO pay to be there.