The effect of “Mist Wurm” targets 1, 2, or 3 cards your opponent controls.
If you use a monster with a mandatory Trigger Effect like “Goblin Zombie” as a Synchro Material Monster for “Mist Wurm” you will activate both the effect of “Goblin Zombie” and the effect of “Mist Wurm” after the Synchro Summon is successful.
Even if there are no cards on the opponent's field, the effect of "Mist Wurm" activates and makes a Chain Link. (Because it has no targets, the effect resolution is not applied.)
If the effect of "Mist Wurm" is activated by targeting 2 or 3 cards, even if 1 of those targets is no longer on the field when the effect resolves, the remaining targets are returned to the hand as normal.
Mentions in Other Rulings
Re-dyce-cle: If Synchro Summoning with the effect of "Re-dyce-cle" activated in the Graveyard, if there are appropriate Synchro Materials for that WIND Synchro Monster, 3 or more Synchro Materials can be used to perform the Synchro Summon. (The total Levels of "Speedroid Red-Eyed Dice", "Speedroid Double Yoyo", and "Speedroid Razorang" are 9, and are the necessary "1 Tuner + 2 or more non-Tuner monsters" Synchro Materials, so can be used to Synchro Summon "Mist Wurm".