If a monster Synchro Summoned with “Torapart” attacks, the opponent cannot activate Trap Cards until after the Damage Step is over. Even cards like “Hero Signal” cannot be activated (which activate when monsters are sent to the Graveyard at the end of the Damage Step).
If a monster Synchro Summoned with “Torapart” attacks, and is destroyed and sent to the Graveyard during the Damage Step, the effect of “Torapart” still prevents Trap Cards from being activated until the Damage Step is over (example: if it attacks a face-down “Man-Eater Bug” and is destroyed by that effect).
The opponent of the player who used “Torapart” for a Synchro Summon is the one who cannot activate Trap Cards. If you take control of your opponent’s “Torapart” and use it for a Synchro Summon, your opponent cannot activate Trap Cards when that monster attacks.
The effect that “Torapart” gives to a Synchro Monster lasts as long as that Synchro Monster remains face-up on the field. If a Synchro Monster that was Summoned with “Torapart” is flipped face-down or leaves the field, the effect of “Torapart” is no longer available, and the opponent can activate Trap Cards when that Synchro Monster attacks.
If “Skill Drain” is active while you use “Torapart” to Synchro Summon a monster, “Torapart” is now no longer on the field, so the effect of “Torapart” will be applied. So the opponent cannot activate Trap Cards when that Synchro Monster attacks. This is because the restriction on activating Trap Cards is an effect of “Torapart,” not an effect of the Synchro Monster. If “Torapart’s” effects are being negated by “Forbidden Chalice,” the same rule applies – Trap Cards cannot be activated when the Synchro Monster attacks.