Is passive perception always on?

I mean, my solution to 5.5e stealth is to use 5e rules -- yes, they're far more dependent on the DM's rulings about when circumstances are appropriate for stealth, but I'd rather have an explicit DM's call than incomprehensible rules.

While taking the Search action let you make a Wisdom (Perception) check, it's not a neccessity. You can make one without taking the Search action.

How often DM ask player characters for a Wisdom (Perception) check to notice something?

Sometimes, asking players to make Wisdom (Perception) checks for their characters tips them off that there's something they should be searching for, giving them a clue you'd rather they didn't have. In those circumstances, use characters' Passive Perception scores instead.