Whether the bullet physically contacts the drops or not they exert a force that affects them. The wind doesn't physically contact a supersonic bullet either but as we've all seen it exerts a great amount of force that affects the bullets flight.
A drop of rain, will impart no linear affect on a bullet in flight. The first issue, getting a bullet and a drop of rain to occupy the same air space at the same time. Next, we need to consider the mass of the bullet v. the drop of rain. The difference in force of the bullet traveling perpendicular to the rain falling at terminal velocity (much less force). The bullet is pushing air, which in turn will "push" the rain out of the way. This lost velocity that we calculate (drop) is realized with, or without rain. That said, there will in fact be a negative affect to velocity. Every action has an opposite and equal reaction type thing. Any way, all of this to say that while the bullet will slow down ever so slightly, it won't actually hit the drop of rain. So yes, rain will affect a bullet, but so negligible as to not even have to calculate for it.
I concede that a bullet with a mass less than a drop of rain, flying at a very slow rate, you will technically be able to start impacting water droplets. I don't know where that point is. BB gun, air soft, I have no clue.
With larger caliber bullets the lower barometric pressure and air density seem if anything to make them run a bit faster than they do on dry days.
100% agreed that air density will effect the flight of a bullet. Easiest explained as how much the air weighs, that the bullet must pass through, or "push" out of the way.
Of course, I could be wet on the whole matter.