Is a "universal" rotor compatible with all types of disc pads?

Every brake manufacturer recommends only using their own rotors, so this is another case where the answer depends very much on who you ask.

In general, here is where rotor cross-compatibility can go wrong:

• Some inexpensive Shimano rotors have less hardening and are designated "resin only." They in fact wear very fast if a metallic or semi-metallic pad is used.
• Magura rotors are 2mm thick and more or less everyone else's are 1.8mm. Using a thicker rotor than a caliper is designed around, along with unworn pads, could get you into situations where you don't have enough air gap to play with even with the pistons all the way in. Such issues come up rarely in practice however.
• Different calipers can place the size and location of the pad contact patch on the rotor a little differently. It seems like in practice there is not a ton of variance, but there is some. Someone trying to design a universal rotor would be trying to make it big enough that the top of the pad on any caliper never overhangs the rotor, but the top of the rotor can never interfere with any caliper.

The main question with the idea of a universal rotor design is whether, given an on-spec location of the brake mounts, is there a magic number for the exact outside diameter needed in the last point. I don't know the answer to that. But for practical purposes, all of the above issues are rare, most rotors are designed to capture some aftermarket sales as generic repair parts, and it's all pretty cross-compatible as long as it's not resin-only or Magura.