Yes there is notation just for drums. I played drums and percussion (different) for the last 40 years or so, on and off. No mostly off.

Each line of the scale denotes a different drum. x is cymbal. For tympany you might have the usual two, tuned and that is using the scale, the music is more like regular notation. You might re-tune during a piece, or have more kettle drums.

Whatever might be considered a "standard" drum set could be a whole long debate. Most music is written for the snare, 3 toms, 2-3 cymbals, high-hat and Bass drum.

How someone would write for 13 toms, for example, would be a seperate notes for each drum. As if each drum was a different note.

There are instances of tuned bongos or drums, playing melodic parts. Very unusual. With the era of electronics and sampling, you could have a whole keyboard of drums, one for each note, or a different sound for each key. Pretty much unlimited.

Toms are not usually tuned to a particular note, and snares can vary according to size and depth. Everything from a so called Jazz slimline to a marching drum.

For percussion scores, each instrument, would have it's own line. For example if it was six different instruments, cymbal, wood block, triangle, timbales, bongos, cow bell... you might have only the sheet music, for ones own instrument, while the conductors score would have all.

As usual, drums aren't like anything else.

Q. How do you get a professional Drummer off your porch?
A. Pay for the pizza (rim shot here)