The second picture is balance staffs. they are printed or or noted with a supplier assortment number, a maker and part number, as well sizes.
Be aware that in 99% of the cases these parts are not rare or expensive, so you could as well group them by maker, in a lot of 4 or 5, include all the writings in the description, take a group picture and let the buyer figure if what they need is in there.
Long story short I serviced a ladies swiss watch with a generic-nothing-special movement.
It wasn't running before but looked like it needed just a service. No damaged parts as far as I can tell but there was some rust around mostly in the keyless works area.
The watch seems to run but it stops, sometimes, if I change the angle or lightly shake it. Also the hairspring doesn't seem to spin very energetic, seems just a tad lazy. If I remove the incabloc rubies system it feels a bit better but haven't tried changing the angle...
One thing that I did not check or clean is the mainspring.
With the watch unwound if I spin the hairspring around it doesn't seem to stop if I change the angle.
I had the same problem on a Longines and it's what Marc described. It's that little spring that presses on the pinion or whatever it's called that the seconds hand is attached to on the other end.
On mine I had simply misadjusted it.