>>In the TV show, Ghost Whisperer, we're now seeing an extension from the use of "I" as an indirect object pronoun to its use as a direct object pronoun.

That's a bit problematic to start with, as "I" is always a subject pronoun, never direct or indirect object (that's "me").

>>I can't remember exactly what the show's character, Melanie, said, but it was something like "Do you want Jim and I to do it?"

This appears to have started in the USA. I first became aware of it years ago in the expression "between you and I" but it's spreading, both geographically, beyond the USA, and into other parts of the language.

Yes, that's not correct in the strict grammatical sense as we should have "me" as an object pronoun; but note that both expressions have "and" -two nouns coordinated, not just one pronoun.

>>I have yet to hear anyone say something like "Do you want I to do it?", any more than we hear people say "Did you give it to I?" I suspect that what we're really seeing is a move towards using subject form pronouns in object positions only when linked by conjunctions, or perhaps only with the conjunction "and".

Yes, good observation.

>>Add to that the use of object form pronouns in subject positions, "Me and Fred went to the shops" - which is similar to something I heard Delta Goodrem say in an interview - and it's becoming a total mess. Again, I haven't heard anyone say something like "Me went to the shops".

Yes, but here I don't think that "me" is to be taken as an object pronoun, rather as an emphatic pronoun (a bit like "moi" in French); it's colloquial, but people have been saying that in spoken language for ages. Typically, their mothers correct them and model what the Queen says ("my husband and I...").

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>>So here's my hypothesis as to what the grammatical rule is becoming:

>>a) In the subject position, the subject form pronoun is used unless the pronoun is linked by a conjunction, in which case the object form pronoun is used.

I'd say the emphatic form of the pronoun is used (instead of the object form) in spoken colloquial English typically in conjunction (but you could also hear "Me, I went to the movies last night")

>>b) In the object and indirect object position, the object form pronoun is used unless the pronoun is linked by a conjunction, in which case the subject form pronoun is used.

Doesn't sound like a consistent rule! Here I think that people say e.g. "Do you want Jim and I to do it" because they are hypercorrecting themselves. Think of my example of mothers correcting children and using the very upper class form when using the pronoun in a subject position. In order to sound more 'correct' speakers may generalise (wrongly) the use of "I" to a number of constructions that remind them of "My husband and I...", i.e. conjunctions. The sociolinguist William Labov has studied language variation and found that hypercorrection did play a part in language change (e.g. lower middle class speakers using a feature they know upper middle class speakers use, and ending up using it more often!). I'm not sure that in this case the practice will spread much further, but it's an interesting example of the principle!

>>Future grammarians are going to have a field day with that.

I think this may have been noticed by linguists before, but maybe not publicised.