1) Yes, projectiles have health. They also have their own "health type", meaning the damage type modifiers are unique for projectiles.
2) With the exception of the armed mine, projectile hitboxes are quite small and moving fast enough for hitting them to be problematic for the physics engine (we already have special-case logic to prevent them from passing through targets that are much larger and slower-moving)
So, yes, it's in theory possible to shoot down projectiles but I've never seen it mean much in practice. It's been this way basically forever, too; some changes to hitmarker display just meant you could start seeing them relatively recently. It was a mechanic we experimented with early on but never found to work in practice - since it also never seemed to be causing problems in practice, no one's ever done the work to remove it.