Cast a Spell does not inherently trigger Attacks of Opportunity.
Cast a Spell, on its own, is not a Manipulate action (it does not have the manipulate trait), it is not inherently a ranged attack, it is not a move action (it does not have the move trait), therefore it does not provoke attacks of opportunity on its own.
But Components can change that
No, you can't.
Your reasoning is correct. Activating the Tentacle Rod isn't an attack, so it can't be an opportunity attack. (The Rod itself makes attacks, but only when you activate it--it doesn't get its own turns, so it can't take reactions of its own.)
If you really want to use it reactively, you can ready an action to activate it, but that's at the ...
You cannot use the Tentacle Rod as it requires an action to activate
The "Use an Object" section of the PHB states:
When an object requires your action for its use, you take the Use an Object action.1
Thus, activating the Tentacle Rod requires your action and opportunity attacks do not allow you to take actions, only reactions.
This is similar to the ...
Cast a Spell can sometimes trigger an Attack of Opportunity
The Cast a Spell action can sometimes have the manipulate trait, depending on the components needed to cast the spell:
The spell components, described in detail below, add traits and requirements to the Cast a Spell activity. If you can’t provide the components, you fail to Cast the Spell.
RAW, you cannot make Attacks of Opportunity with Automatic Firearms.
An Attack of Opportunity is a single attack that targets one creature. The Automatic quality disallows this form of attack. If you are wielding an Automatic firearm, you cannot make the type of action necessary for an AoO, so you cannot AoO, even though you still threaten with Snap Shot.