In general, bonuses of the same type do not stack. This means that all allies will only get the highest single morale bonus for the circumstance between Banner and Flagbearer. Since Banner of the Ancient Kings doubles the bonus from Flagbearer, the Banner bonuses only apply to allies outside the 30ft range of Flagbearer but that are within the 60ft range of Banner.

The results would be:

+4 total against fear and charm: +2 morale from Flagbearer (+1 x2 from BotAK), +2 resistance from Banner of Ancient Kings

+2 against other mind-affecting: +2 resistance from Banner of Ancient Kings

+2 bonus on attack rolls: Flagbearer grants a +2 morale bonus on attack rolls (+1 x2 from BotAK)

+2 damage on attack rolls: Flagebearer bonus x2 (+1 x2 from BotAK)

+4 on initiative: +4 circumstance bonus on Banner of Ancient Kings

As long as you wield the spear in two hands, I don't see why the bonuses are gone as soon as you attack -- they should remain and apply to allies as long as they can see the banner. On the other hand, if you use a shortspear, you can still attack by wielding the weapon in two hands but you can't attach your banner of the ancient kings to it ("As long as the longspear or pole to which the banner is attached is wielded in two hands...").

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.