Few questions on hybrids and genetics

Thank you for reading. So I have a basic understanding of Mendel's works, but I am trying to figure genetics out better. Please if you could help, because I can't find the data anywhere on the internet on these questions.

If you take an F1 hybrid and cross it to a p1 (seperate from the hybrid's two different parent types) would you get a new F1 hybrid, or F2?

If you take and F1 hybrid and cross it to a seperate type of F1 hybrid, do you get a new F1 hybrid of F2?

F1 hybrids are said to be uniform, is there any exception that a seperate phenotype can pop up even very rarely in a F1 hybrid?

Would you use the term BX1 or BC1 for backcrossing, or are both acceptable terms in the scientific community?