1. PC invisible (via dust of disappearance). Moves hex spell to a new target. Lose invisibility?
No. Invisibility specifically states, "The spell ends for a target that attacks or casts a spell." You're doing neither here.
2. Bonus action spell already cast. Activate magic item to cast fog cloud (magical bard instrument). Violates bonus action spell/only one action cantrips rule?
It depends on how you interpret the rule for using an item. Personally, I think magic items should be special. Technically, since you're using an action to play an instrument that happens to cast a spell, I'd allow it. However, another DM might rule otherwise given that the item description does say, "You can use an action to play instrument and cast one of its spells."