Nothing prevents a staggered character using swift or immediate actions
As you've observed, the wording of the staggered condition does not preclude a character using swift, immediate, or free actions when appropriate - or from making attacks of opportunity.
It would be safe to assume that if a character cannot act at all they cannot take such actions. Indeed, 3e's description of conditions relies on a certain amount of common sense interpretation, since the descriptions for being helpless or unconscious don't explicitly say you cannot take actions either, but we reasonably understand that unconscious characters do not act.
However, that is not the case for a staggered character, who is restricted to taking fewer actions than they would normally be able to, but is not prevented from acting entirely. The condition indicates that the creature finds acting difficult, but not impossible. Contrast to the stunned condition, which explicitly states that the character "can’t take actions" - and that would mean any kind of action, including swift/immediate/free actions.
It's worth noting that the concept of swift and immediate actions arose relatively late in 3.5e's lifetime and so the wording of material written prior to their introduction never explicitly considers them. When Paizo was reworking the system for Pathfinder, they rewrote the staggered condition to reflect the wider system of actions:
A staggered creature may take a single move action or standard action each round (but not both, nor can he take full-round actions). A staggered creature can still take free, swift, and immediate actions.
I'm not aware of anyone who thinks that's a change to the rules as intended in 3e.