You'll find them both ways which are considered correct. I like the looks of the headless cocking piece but just because it is different than normal.

Not something to worry about. there are often other issues more important like matching patina on the metal and stock and condition of the bore. The rear sight is a big concern about correctness, also for carbines especially.

Hopefully there is nothing altered from original military condition with the stock or barrel or front sight. All Model 1899s were carbines originally.

If you can post pictures, we'd love to see what you have. Welcome!