You could put your propellers outside and away from the cage, with their motors inside the cage. This would also greatly reduce the size and weight of the cage.
Heck, all the wires, electronics, and motors could be inside the "arm". Then coat that with some EM shield material (like a metal foil).
I like it.
Of course, from "like it" to "it's actually effective" there's still the nagging little field test to perform. It's the old shell-vs-armor game, since EM radiation has a penetration depth - Faraday cages are not magically 100% effective, make a thin enough one, and a powerful EMP would still do some damage inside. So now you need to carry thicker, heavier EMP armor.
However it also means the drone has no GPS; no wireless signaling; no sensors outside the foil at all. It becomes a dumbfire dead-reckoning device.
No, they are not. Even a continuous, zero-hole metallic bubble will leak some radiation inside, depending on frequency, nature of material, wall thickness, etc.
Start here:
https://en.wikipedia.org/wiki/Penetration_depth
then continue reading about absorbance, skin effect, etc.