Multi channel variable speaker design problem
Yes, it is allowed to have that variable a load on your amplifiers, given certain precautions. Most amplifiers can tolerate open circuit on their outputs, and, subject to limiting the output so they don't get too hot, somewhat lower than their 'rated' load.

A common method for supplying multiple speakers from a central location is the '100v line' method. Each speaker has a transformer before it to raise its impedance. This can get expensive in transformers, that have to be rated for the audio range.

You should be able to buy high impedance speakers, 32 ohm is not uncommon. Ten of those in parallel would present around 3ohms to your amplifier, which most audio amps should be able to drive at less than max output with little distress. How much fi and power do you want, I suspect these higher impedance speakers will be lo-fi and low power.

You could precede each speaker by its own local amplifier, which would raise the per room loading to a level where it ceases to be an issue. If powering the amplifiers via the CAT5, you would want a beefy electrolytic across the supply at the amplifier.