Elemental body places no restriction on spellcasting
Nowhere in the elemental body spell is any limitation on spellcasting offered. It simply says you take the forms of the various elementals.
Elementals are capable of casting spells
Unlike animals, elementals can talk, tend to be roughly humanoid in appearance, and thus can perform gestures and manipulate materials and foci. Elementals can take levels in spellcasting classes, and when you become one you remain capable of doing everything that’s necessary for spellcasting.
So yes, you can cast spells during elemental body.
All your items meld during a polymorph effect
This would include your spell components, which is problematic for spellcasting. A sorcerer, of course, has Eschew Materials, and anyone else could take it, so that would be one solution for negligible-cost components, or you could just keep the components separate while polymorphing.
Considering how easy it is to get around, and how it really seems like it’s supposed to be a convenience feature for the transmogrifier, I probably wouldn’t bother enforcing this rule though. I could see abuses that might make me put my foot down, but for the most part it’s probably fine.
Probably yes, a sorcerer with an Intelligence score of 2 yet possessing sufficient Charisma can still cast spells with verbal components
The GM can rule that a low-Intelligence sorcerer can't cast spells, but the game doesn't outright say such a sorcerer can't cast his spells. Therefore, ultimately, the GM must decide. That's because according to Components
A verbal component is a spoken incantation. To provide a verbal component, you must be able to speak in a strong voice.
While Pathfinder has several sets of rules for things called incantations (e.g. here, here), the term incantation as mentioned in Components is undefined, so the GM can then consult his favorite dictionary and use that definition of incantation to tell the sorcerer player No spellcasting for you.
Further, the GM may rule that the sorcerer's low Intelligence score makes him incapable of verbalizing at all, dropping the poor sorcerer back to the prelinguistic stage of language development.
But, assuming that the GM doesn't want to player to head off to the other room and play Smash Bros. until the sorcerer's Intelligence is at least 3 and that the player appreciates the challenge of role-playing, essentially, a dog with a built-in shotgun, the DM should permit the low-Intelligence sorcerer to cast spells.
Note that I would be very wary around a dog with a built-in shotgun, especially if that shotgun were reloaded automatically each time the dog wakes up from a long nap. I'd go to great lengths to keep that dog fed, happy, safe, and appropriately amused.
Because it's been referenced in several Comments, in Pathfinder's antecedent Dungeons and Dragons 3.5, under Monsters as Races it says
Creatures who have an Intelligence score of 2 or lower, who have no way to communicate, or who are so different from other PCs that they disrupt the campaign should not be used.
Then, later under Ability Scores for Monster PCs, it says
The separate table for [modifying a PC's] Intelligence [score when the PC is a monster] ensures that no PC ends up with an Intelligence score lower than 3. This is important, because creatures with an Intelligence score lower than 3 are not playable characters.
Neither artifact, so far as I can tell, appears in Pathfinder.
Best Answer
On Special Abilities on Spell Resistance says, "A creature’s spell resistance [SR] never interferes with its own spells, items, or abilities." Thus, for example, if the boss were to cast a silence spell on herself or an object, the boss's own SR can't "interfere" with how that spell functions. That means—and, in the boss's case, this is unfortunate—that the boss's own silence spell works normally, so it also mutes the boss, therefore typically preventing the boss from uttering verbal components.
However, if another creature were to use a silence spell, then the boss may be able to ignore the silence spell. That is, if a creature that possesses SR enters the area of an ongoing spell and the spell's caster fails to overcome the creature's spell resistance, then the creature's spell resistance "protects the [spell] resistant creature without affecting the spell itself."
This is later further clarified: "Against an ongoing spell that has already been cast [like a silence spell upon an object], a failed check [by the spell's caster] against [the spell resistant creature's] spell resistance allows the [spell] resistant creature to ignore any effect the spell might have. The magic continues to affect others normally."
So while the boss's strategy won't work if the boss does it personally, getting a minion to help out is legit.