This is an interesting question, that I have also been thinking about for a long time. But in my thinking, a lot of importance goes into making the center of thrust agree with the center of mass, a point that many copter builders seem to totally forget!
Please correct me if I’m wrong, but I believe that the “pendular stability” achieved by having the center of mass lower than the center of thrust does not help at all in actively/automatically controlled aircraft, because all the required stability is provided actively, by the combination of various sensors and fast data processing, that uses several configurable parameters to optimize stability. We just don’t need to add passive stability. Whether we place our propellers above the center of mass, or at the same height, or even below it, changing pendular stability from positive to neutral and even negative, makes no change in actual stability, after the parameters in the software have been properly set. Right?
But there is an important other point: Power distribution between motors. If the center of mass and center of thrust are coincident (in all three dimensions), then during stable flight conditions (hoovering, or flying at speed, with or without wind) all motors will be required to produce equal thrust, and thus run at the same power. Power changes occur only briefly, while changing the craft’s attitude, or accelerating it.
If instead the center of thrust is higher or lower than the center of mass, then whenever the copter is in an inclined position, half of the motors need to continuously produce more thrust than the other half. This condition limits the thrust reserve for manoeuvering, the cargo capability, and it also reduces efficiency, and thus flight time. Particularly flight time is affected, because a higher final battery voltage is required to maintain adequate manoeuvering reserve when the craft is imbalanced in this (or any) way.
When flying fast enough, aerodynamic effects come into play too, but I believe that due to the mentioned thrust distribution between motors, it should always be best to place the propeller plane so that it intersects the center of mass.
From a practical, operational point of view, I dislike having the propellers face down, with nothing below them, because it’s a fact of life that some landings happen with the copter not totally leveled, or with some residual horizontal speed, and I just don’t enjoy landing on my prop tips…
So I think that an optimal configuration when designing a copter is placing the center of mass as high as possible (battery on top), then place the props as close as possible to the height of the resulting center of mass, and then place the arms as low down as possible, under the props, using the arms as landing gear. That way the arms are as far away as possible from the props, minimizing interaction, the motors can all work at equal thrust even when the copter is inclined, and the structure of the frame is simple (needs no separate landing gear).
Of course the arms should be as narrow, smooth, and aerodynamically shaped as possible, consistent with stiffness, to further reduce aerodynamic interaction with the props. But the importance of this might not be very high.
About vibration: In my own, admittedly very limited experience, the largest contributor to vibration is mass imbalance, rather than aerodynamic interference with the arms. I see that very few people dynamically balance their motors and their props. At best they do a static balancing of their props, which is definitely not sufficient to achieve low vibrations! I invested a lot of effort into doing a sort-of dynamic balancing of each of my motor+prop combos, fully assembled. All it takes is loosening the prop and retightening it, and the balance is gone, the vibrations are back! So it’s of little use to balance a propeller by itself, then mount it on a motor. When I get the balance just right, after much trial and error and many little pieces of tape, small spots of hot glue, layers of varnish at various places, the vibrations are very low. If I then wrap my copter arms in something that drastically changes their aerodynamic behaviour, not much changes with the vibrations. But all it takes is a prop catching a blade of grass, and the balance is gone, the vibrations shoot up.
Surely this is not the same in all copters. Depending on prop size, type, speed, arm shape, size, surface, the interference effects on vibration might be stronger in some copters, and vibration might be more or less of a problem, but in mine (a good(?) old F-450), mass imbalance dominates by far. Which makes me think that the vibration advantages of placing the props under the arms instead of above them might be irrelevant in most cases.
A lot of vibration can also come from aerodynamically imbalanced props. For example, think about a 2-blade prop that has a slightly crooked seating plane, so that one blade ends up with a higher angle of incidence than the other. The result much more thrust from one blade than from the other, and fierce vibration even if static and dynamic mass balance is perfect.
I went through dozens of props of different brands and types, selecting the least crooked ones, rectifying their seating planes, enlarging their center holes to make them straight and well centered, then inserting precise bushings with a ring-shaped contact to the shaft, to remove play without crooking them again… And it all helps a lot, but all it takes is a slight hit against something, and the propeller will again be out of balance and alignment.
So what I suggest is: Don’t see it as “props over arms” or “props under arms”, but rather see it as “center of thrust coincident with center of mass”, and then place the arms wherever it’s more practical for your intended use. In most cases that’s arms below propellers. Keep the arms reasonably thin and smooth. And do watch the static and dynamic balance of the complete, assembled motor+prop combos, and the aerodynamic balance of the props!