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.
No.
A Barbarian does not benefit from the special damage resistance from the bear totem while in heavy armor.
...but the 6th-level totem choices can benefit a Barbarian wearing Heavy Armor if it doesn't mention that they can't be wearing such armor to gain the benefit.
According to Jeremy Crawford: The intent is that anything that relies on Rage does not benefit a Barbarian wearing Heavy Armor.
Why? I take this ruling to mean a Barbarian is incapable of gaining any and all benefits of Rage while wearing heavy armor, totems included.
To me, this means you simply can't Rage while wearing heavy armor, because there would never be any usefulness or benefit to entering a Heavy Armor Rage; however, Crawford has added more context to his ruling that states that Heavy Armor does not prevent you from entering a rage, only benefiting from it.
As you quoted, it says you gain the benefits of Rage only if not wearing heavy armor... so if you are wearing heavy armor but can not benefit from Rage, you also do not benefit from any features that depend on Rage, such as the 3rd level totem features.
Best Answer
Alas, there is no minimum damage value. From page 10 of the Sage Advice Compendium:
Therefore, due to the Round Down rule, the raging barbarian wouldn't be immune in the sense of game mechanic terminology but they would take 0 damage from something that only does 1 point of damage of a type that the barbarian is resistant to.