Would seasonal trees last all year if the weather permitted it?

Nope. Deciduous trees will still go through an annual cycle in mild weather.
In Southern California people plant and grow everything, there’d be a heap of leaves on the ground every October from the leaf-droppers.

One year I drove to Sycamore Canyon, just north of Los Angeles, to see if the leaves fell. They did.

http://www.venturacountytrails.org/TrailMaps/Sycamore/AreaTrails.htm

The trees are programmed to drop their leaves.

Is the season for dropping leaves the opposite in the northern and southern hemispheres? Are there deciduous trees in rain forest climates? If cold snaps cause leaves (example, maple) to change color and then fall off, what signals leaves to fall off if there isn’t a cold snap ever?

We have trees that don’t drop all their leaves, it is the oddest thing. They all go into hybernation, the leaves dry up and go brown, but an unusual amount don’t fall on many of the trees. Don’t get me wrong, it is still the majority of the trees that lose all their leaves, but the ones that don’t is a high enough percentage that it is noticed. This happens even during cold winters, although very windy autumns, or snowy winters, help bring down the leaves. In the spring when the trees get buds, the rest of the old leaves fall down.

@JLeslie Oaks tend to hang on to their dead leaves for a long time. In some varieties, the old leaf won’t drop until the new leaf pushes them out.

Deciduous trees excrete waste to the leaves which, in turn, are carried away by the breeze and gravity. If you stop excreting, you have trouble. (I know this from horrid personal experience.)

Trees are controlled by sunlight, especially red light, not weather.