Learn English – Because me and the family or because I and the family

You are correct: me is the object pronoun and should be used here, since it is going in the object position.

Normally you would probably say "keep us updated", so "keep him and me updated" may feel a bit off even though it's perfectly correct.

As to the people who tell you that you should always use "I" in conjunction with a name: they are WRONG, WRONG, WRONG! Ignore them! It's a form of "hypercorrection"--when you "correct" something that's already right based on some rule that doesn't actually apply.

Too many people have gone from:

"you never say 'Tom and me went to the store' (because "Tom and me" is the subject, so you need I instead of me)"

to

"You never say 'Tom and me', ever"

and then had to invent a reason for their mis-remembered rule.

My parents will reside with me and my family.

This sentence is okay, because if we shorten it, leaving only me and omitting "my familiy", we notice nothing strange:

My parents will reside with me.

Because this pronoun is not in subject position, it's okay to use the form me. The pronoun is the object of the preposition with. Such pronouns are called object pronouns.

Your second sentence is actually considered ungrammatical (but look below for a fuller view):

My parents will reside with my family and I.

Why? Because I is a subject pronoun, and should not be used in object position. Let's see how the sentence will look if we omit "my family":

My parents will reside with I.

This is a strange-sounding sentence.

In "standard" English there's a tradition to frown upon the use of me in subject position. Why? Because when we omit the "and-part", we get this:

Me will reside with my parents.

Sounds a bit strange. The "correct" form of the pronoun for subject position is I.

P.S. When someone attempts to use I instead of me in object position in order to be more "formal", it is called "hypercorrection". Quoting Wikipedia,

Jack Lynch, assistant professor of English at Rutgers University, says that correction of "me and you" to "you and I" as subject leads people to "internalize the rule that 'you and I' is somehow more proper, and they end up using it in places where they shouldn't – such as 'he gave it to you and I' when it should be 'he gave it to you and me.'"[7]

On the other hand, the linguists Rodney Huddleston and Geoffrey K. Pullum claim that utterances such as "They invited Sandy and I" are "heard constantly in the conversation of people whose status as speakers of Standard English is clear"; and that "Those who condemn it simply assume that the case of a pronoun in a coordination must be the same as when it stands alone. Actual usage is in conflict with this assumption."[8]

That's a nice observation by H&P. My test with leaving out "my family" and dissolving the coordination is invalid, from this point of view.

So let's say that's basically how the awkwardness of I in object position and of me is subject position is usually explained. Linguists are arguing about whether it is really ungrammatical. "The jury is still out", as they say.

The take-home message is that your sentence 2 will be marked off as erroneous if you take an English language test.

Related posts:

• Is it "Tom and I" or "Tom and me" in "Keep Tom and I/me updated"? -- the deepest answer is in this post by F.E.. A definitive answer, I might say.

• Can ME be a subject? - a nice answer by Snailboat to a related question.

• Between you and me - or Between you and I - on ELU SE.

• “My friends and I” vs. “My friends and me” vs. “Me and my friends” - on ELU SE - with a great psychological explanation of why we tend to use me in coordination.