OK, so here’s the text in question:
Spell Rage (Ex)
A rage mage can cast spells while in a rage, as long as the spell’s casting time is no more than 1 full round. When she casts a spell of the abjuration, conjuration, evocation, necromancy, or transmutation school while in a rage, the rage mage uses her character level as her caster level. This ability only works when the rage mage is in a spell rage.
While in a spell rage, a rage mage becomes reckless and loses her some of her ability to defend herself. She temporarily takes a –2 penalty to Armor Class. She can attempt Concentration checks to avoid having a spell disrupted, but can’t use any other Charisma-, Dexterity-, or Intelligence-based skills (except for Balance, Escape Artist, Intimidate, and Ride) while in a spell rage.
A fit of rage lasts for a number of rounds equal to 3 + the character’s Constitution modifier. A rage mage can end her rage voluntarily. At the end of the rage, the character is fatigued (–2 Strength, –2 Dexterity, can’t charge or run) for the duration of the current encounter (unless the rage mage is 10th level, when this limitation no longer applies). A rage mage can only fly into a rage once per encounter, and only a certain number of times per day (determined by level). Entering a rage takes no time itself, but a rage mage can only do it during her action, not in response to somebody else’s action.
This is a bit ambiguous. It mostly refers to Rage, not Spell Rage. From the text, it could be read as a change to all Rages. However, the table indicates “Spell Rage 1/day” and then later “Rage +1 use/day.” This indicates that they are separate features with a separate pool of uses.
Effectively, Spell Rage is a separate effect that has to be activated separately from Rage, and uses a different pool of uses. Since both are activated as free actions, you can always activate both at once to gain the benefits of each. Note that Spell Rage specifically allows you to cast spells during a normal Rage, overwriting the usual restriction in Rage that prevents that.
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.
The effects do not stack
We can read about Combining Game Effects in the DMG errata, expanding DMG page 252:
Here, the effects are the same and obviously have the same name. Thus the effects do not stack. However, they do not cancel each other, so both will be active at the same time. Using this the barbarian could effectively extended the duration by raging again, as when the first rage ends, the other will still be going on, granting him the bonuses.
Note that only the damage bonus could even stack, as the other bonuses grant Advantage, Disadvantage or Resistances, which do not stack with themselves, regardless of the feature granting them.
I do not recommend overruling the no stacking rule, as that will be a problem at level 20 and a bit for Zealots. Stacking the damage bonuses indefinitely at level 20 could lead to ridiculous and imbalancing results. It is not as bad for not-20 Zealots, but still an issue I think.
Overall, I do not think this feature was designed with this option in mind and would recommend that DM-s rule that it does not stack. I would even say that ruling that you cannot start raging while already doing so would be quite reasonable and would avoid potential problems.
(Thanks to V2Blast for pointing me to the Errata.)