When a Trap Card that inflicts damage is activated, this card is activated directly in Chain to that activation.
Resolving "Negate the activation of that card" and "destroy it and all face-up Spell/Trap Cards your opponent controls" are treated as being done at the same time.
Whether or not "The Paths of Destiny" is a card that restores LP or inflicts damage is determined by a coin toss performed when the card resolves. "Shining Silver Force" cannot be activated in Chain to the activation of a Trap Card that may or may not inflict damage.
After "Skull Invitation" resolves, while it is face-up in the Spell & Trap Zone, the effect "Each time a card(s) is sent to the Graveyard, inflict 300 damage to its owner for each card sent" is applied. Since Skull Invitation does not inflict any damage when the activation of the card resolves, the card "Shining Silver Force" cannot be activated in Chain.