I have a character designed that is two different characters that have been merged into one body by some force (magic, a god, whatever) So they share one physical body, but they have two fully separate minds. Are there any published rules that might cover this situation?
I don't know of a canonical rules source for "here's how it works in D&D 3.5 when you have two characters in a single body," but I can point you to a couple of similar cases for which there are rules, from which you might draw some inspiration.
The psionic power Schism splits its user's mind in two, with each half able to take its own actions. While the source of the split mind for your character (presumably) wouldn't be the same, the description of Schism's effects sound a lot like what you're describing:
Your mind splits into two independent parts. Each part functions in complete autonomy, like two characters in one body. Your new “second mind” does not control your body physically but is free to take one standard action in each round if the action is purely mental (such as manifesting a power) in the same round you take your normal actions.
If your character concept concept is consistent with this setup, where one mind has control of the body and the other can only take mental actions, you could borrow from Schism's mechanics. Of course, since this option gives you additional actions per round, it would be a net increase in character power, so you might want to add some disadvantages to balance it out.
Fiend of Possession
The Fiend of Possession prestige class (Fiend Folio) has the ability to become incorporeal and possess characters and objects. Again, this is different from the fluff you have in mind for your character (where two characters are fused, rather than one forcibly possessing the other), but you can still borrow mechanics.
Again, this results in one of the minds being restricted to mental actions only:
While possessing an object, a fiend of possession can use any ability it has that requires no physical action, such as using a spell-like ability or telepathy. It cannot cast spells (since it can neither speak nor move), attack physically, or perform any other physical action, until it reaches a high enough level to make the possessed object perform these tasks for it.
The Dvati (a race from the Dragon Compendium) are in some ways the opposite of your character concept (they're a single soul split into two bodies that counts as a single character), but you might want to reference them in terms of what kinds of actions might be restricted for action economy purposes.
For instance, when a Dvati character casts a spell, both halves are required to spend their action to do so (because doubling the number of spells a character can cast per round is ridiculously powerful), but they can move independently, each spending only their own action.