If I put a new sponge filter into an already established cycled tank, how long will it take for that sponge filter to become cycled itself?
And then how long might it take for a cycled sponge filter to jump start and fully cycle a new uncycled tank?
I would run it in a cycled tank for at least a month before I would use it to jump start or instantly cycle another tank. The amount of bacteria on the sponge and its ability to help cycle another tank is going to depend on the bio-load in the tank you are seeding the sponge in.
I have used well seeded sponge filters from a heavily stocked tank to instantly cycle another tank. As long as the bio-load is lower in the new tank than it is in the tank the sponge came from there will be enough bacteria to handle the bio-load of the new tank. That is what a cycled tank does.
It is still going to take a while for the new tank to get established but you should not get either an ammonia or nitrite spike in the new tank. You should start seeing nitrates within a week or so.