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.

A barbarian/bard can, while in a rage, start and continue a bardic performance—maybe

The bard's bardic performances say nothing about concentrating and only a handful mandate Perform skill checks (countersong, distraction, et al.), making the others possibly usable while in a rage. However, performances do require using some kind of action to start and, often, to maintain. So while it's totally legit to get really angry and, for example, climb a mountain or swim a channel, because bardic performances use actions, starting or maintaining them while in a rage might run afoul of this part of the rage description:

While in rage, a barbarian cannot use any Charisma-, Dexterity-, or Intelligence-based skills (except Acrobatics, Fly, Intimidate, and Ride) or any ability that requires patience or concentration.

Emphasis mine. So while the barbarian's rage itself doesn't explicitly forbid, for example, massive shredding on your lute or belting out "99 Bottles of Dwarven Ale on the Wall" while simultaneously furiously headbutting orcs to death, the GM can simply say No, starting and maintaining a bardic performance requires patience that's impossible while raging.

This GM would allow a character in a rage to start or continue an appropriate bardic performance

The player should be aware that it's a tough row to hoe, though. In addition to other issues like multiple ability score dependency,¹ the huge—perhaps, I dare say, insurmountable—problem with playing, for example, a barbarian 3/bard 4 is that such a character will each day have only a total of 11 rounds of fight in him (and that's generously assuming a Con 16 and Cha 16).² Level 7 is actually past the point when the wizard can cast an extended rope trick and everybody can rest in the extradimensional space in relative safety, so being good for only two fights per day (unless the group's really efficient) is fine at that point, but actually playing this character to that point would be a constant and—for me, anyway,—unpleasant war with an ever-ticking clock.

¹ Such a character needs high Str, Con, and Cha, would like a high Dex and Int, and will regret a low Wis.
² Yes, I'd put the extra level in bard. That means 2nd-level bard spells.