Due to the golden rule, only teleportation-based forced movement effects can move the minion out of the Bullfrog's belly; non-teleportation, forced movement effects can't move the swallowed minion away from the Bullfrog. For example, Blink or Swap would allow the minion to escape, but River Rapids, Riptide, or Scatter would not.

What happens if the Bullfrog is disabled or silenced while carrying a minion?

The Bullfrog's Genesis ability creates an ongoing effect which allows the Bullfrog to continue to carry a minion in its belly, though it cannot Drop that minion or Pick Up another one. Because this comes from an ongoing effect, the Bullfrog being disabled or silenced will not cause the minion to be dropped.

If the swallowed minion is teleported away somehow, does it lose Disabled?

Yes.

Can the Bullfrog choose to drop the minion it swallowed?

No, it can't choose to drop the swallowed minion. (If the swallowed minion is teleported away, it will still be automatically dropped.)