No, Bardic Inspiration can't cause a critical hit
Cutting Words states, "...subtract the number rolled from the creature's roll.", so the number is being applied to actually change what you rolled on the d20.
Bardic Inspiration states, "add the number rolled to one ability check, attack roll, or saving throw", so the number is just another bonus on the attack roll.
The difference in phrasing is very small but important here. Terms like, "attack roll", "saving throw", "ability check", are common phrases in 5e used throughout the book. PHB, p.195, explains that you make an attack roll by rolling a d20 and adding appropriate modifiers. If it weren't for the Jeremy Crawford tweet it would be sensible to assume that the, "roll", in Cutting Words refers to the, "attack roll", but that tweet exists so instead it refers to the actual roll of the d20. Bardic Inspiration specifically says, "attack roll", so we can be relatively certain that it is just another modifier added to the roll.
The spell Bless uses the same phrasing as Bardic Inspiration, "...and add the number rolled to the attack roll or saving throw", and there is no reason to believe that it grants (or bizarrely prevents) critical hits.
It doesn't help passive anything.
The key word here is Passive Perception. Think of it as the character's AC against stealth. It's there so that the DM doesn't have to ask for a Perception check any time someone is trying to sneak up on the party. To help emphasise the point, consider this line from the inspire ability.
The creature can wait until after it rolls the d20
You don't roll anything for a passive skill, so nothing for the inspire die to boost.
It helps with active rolls
What you can do however, is inspire someone who is actively rolling a Perception check. If your party thinks they need to actively look around you can maximize the roll by doing a few things at once.
First pick the person with the highest Perception skill. Then any other party member can preform the Help action, granting advantage on the roll. Then the bard inspires the character, adding the die into the pool. The end result will be a very high chance at a much higher roll than a normal Passive Perception.
Best Answer
No.
Magic Missile doesn't have either ability check nor attack roll or saving throw it automatically hits, and a damage roll determines the damage. So Magic Missile has no benefit from bardic inspiration.