Can the chain make AOOs? You'd think so since that's the biggest reason a 10' reach matters, but the spell's intent is not clear.
No. It is a hindering spell that only affects one opponent until the caster spends a move action to change targets. The chain does not move of its own accord like a spiritual hammer would. The 10' reach just gives it a larger area it can guard before a target can move out of its range.
Interposing Hand is a 5th level spell, and Chain of Perdition is 3rd level spell. The interposing hand targets one opponent and does not get Attacks of Opportunity. It does not chase the opponent if they withdraw, but otherwise acts each round to physically prevent the target from reaching you. The 7th level spell Grasping Hand also only moves only when directed to move, and says "The hand holds but does not harm creatures that it grapples."
As a lower-level version in this spell tree, Chain of Perdition should function in a similar manner, just weaker. In this case, it will hinder an opponent with a successful a CMB roll rather than produce a greater effect.
Chain of Perdition is stationary unless the caster moves out of range or out of sight of the chain, at which point it automatically returns to the caster. The chain targets one opponent (which you can change within 30 feet as a move action) and then acts each round with a CMB roll to attempt to hamper that opponent. The fact that the chain is a medium item with a 10' reach simply allows you to guard a larger area with it before spending a move-action to relocate it. For example, a hampered opponent cannot escape the Chain of Perdition's reach by taking just a 5' step.
Also, as a medium object that must be summoned into an unoccupied space, I would assume that it occupies that space as a medium object would. Thus, even stationary, it can be used to hinder movement through that space.