"Emma and us" in sentence

The trick in checking the grammar in this kind of sentence is to remove the reference to "Emma and" (or whomever) and see if it still sounds correct.

When you do that, you'll see that the first one is grammatically correct, since you could potentially write "We want us to be part of it." (Although in this case, only if you were using "us" for emphasis, otherwise you'd be more likely to simply say, "We want to be part of it.")

But the second example doesn't work when you do that - you'd never say "This is where us want to stay." So the correct grammar in that instance would be "This is where Emma and we want to stay."

The same applies to me vs. I. Many people seem to think that, for example, "Fred and I" is always correct. However that is not true if you would say "me" - e.g. "This is a problem for Fred and me" which matches "This is a problem for me."

In grammar terms, it's the difference between whether the pronoun is the subject (we/I) or object (us/me) of the phrase.

Hope that helps!