Pathfinder states that each spell of the Enchantment school as well as each Pattern and Phantasm spell in the Illusion school is Mind-Affecting. See Mind-Affecting^D20PFSRD definition and the Magic^D20PFSRD chapter for reference.

Mind-Affecting
Enchantment spells affect the minds of others, influencing or controlling their behavior. A mind-affecting spell works only against creatures with an Intelligence score of 1 or higher.
All enchantments, illusion (patterns), and illusion (phantasms) are mind-affecting.

So it seems more like a definition of the Enchantment school instead of the definition of the Mind-Affecting descriptor.

Protection from Evil, as you pointed out does not mention the Mind-Affecting descriptor, but states that:

[...] the subject immediately receives another saving throw (if one was allowed to begin with) against any spells or effects that possess or exercise mental control over the creature (including enchantment [charm] effects and enchantment [compulsion] effects). [...]

If we accept these two assumptions:

1. The Mind-Affecting definition does not only refers to Enchantment and Illusion spells (and the Necromancy-tied Fear seems to suggest so).
2. An effect with the Mind-Affecting descriptor is eligible to get a second saving throw under a Protection from Evil spell.

Then the answer is Yes.

Note that this answer specifically address your citation of the Fear's Mind-Affecting descriptor. One could also argue that, as long as the Protection from Evil wording is concerned, a Fear actually triggers the abjuration spell's condition outright because - in fact - it exercise influence over the target's course of action.

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.