An ability that triggers "whenever you tap a creature for mana" triggers only if you activate a mana ability of a creature including in its cost. Mana abilities that don't include the 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 ).

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