Just to keep things simple, let's say a "phoenix" is an oscillator in which no cells ever survive.

In which rules can they exist, and which periods can they have?

Since removing survival rules can never affect a phoenix, finding phoenices is equivalent to finding oscillators in rules without survival.

The LifeWiki article claims a proof has been found that no phoenix in Conway's Game of Life (so in B3/S) can have period 3.

In fact, all the phoenices I can find in any rules have even periods. I've found periods 2, 4, 6, 8, 10, and 14 so far.

I've found phoenices whose simplest rules are B2/S, B24/S, B25/S, B3/S, B34/S, B346/S, B35/S, and B36/S.

Then there's the rule B345/S. There are many oscillators in this rule, including periods 18, 22, 24, 26, 30, 32, 34, 38, 40, 42, 62, 66, 72,