An ability that triggers "whenever you tap a creature for mana" triggers only if you activate a mana ability of a creature including {T} in its cost. Mana abilities that don't include the {T} symbol and instead say "Tap an untapped creature you control" or similar won't cause Badgermole Cub's second ability to trigger. Similarly, it won't trigger if you tap a creature to activate a mana ability of another object (even if that mana ability also includes {T}).

Badgermole Cub's last ability is a triggered mana ability. It doesn't use the stack and can't be responded to.