The rules, as written, are garbage for this feat. They are extremely poorly-written: awkward and unclear.
Therefore, my answer to this question is: Ignore the rules as written entirely. They are a meaningless mess of gibberish that almost, but not quite, means something. And I say this as (as of this writing) the top answerer of rules-as-written questions on this site.
The concept is sound. You can read the feat and get a pretty good idea for a neat feat. But the answers to your questions should all be determined by the DM for his own, homebrew feat, because this feat, as written, doesn’t clearly answer them.
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.
Best Answer
7th level bards
All bards, from 7th level on, can start a new performance as a Move Action. From then on, you can either use a standard or a move action to start a performance, and thus start two different performances in a single round.
At 13th, you should be able to start up to three different performances in a round, using your standard, move and swift actions.
Do note that you can only have one bardic performance in effect at any given time. So, normally, you couldn't have Inspire Courage and Inspire Competence active simultaneously, as you have to stop one performance before you start another.
However, you are not in fact using two performances simultaneously, as the Sound Striker's Weird Words ability is not a bardic performance that lasts for a number of rounds. It is a supernatural ability that "consumes" your daily uses of bardic performance, create a magical effect, and dissipates. In other words, the Weird Word performance is like an instantaneous effect, it begins and ends immediately.
Just remember that Weird Words is "always a standard action" to use.
Mythic Bards
Mythic bard characters who take the Marshal mythic path may learn the Lightning Performance ability, which allows them to start a new performance as a Swift Action instead of a Standard Action:
Using this ability would allow you to start two performances in the same round before you are 7th level, one using your standard action and another using your swift action, as mythic paths follow a different progression from normal character advancement. Meaning that you could be a first level bard with a mythic tier 1 and this ability to start performances as a swift action.