I imagine it has to do with the fact that you are emphasising the "we" because you are contrasting - "(not only others but) we, too" (mi is).
If you had odasietünk is, it might be something like "We (not only eat breakfast, but) also hurry there".
Yes, that's correct. Is refers to the subject, so mi has to be present. Is can't belong to an omitted word.
But if is belongs to a verb with a prefix, it separates them: oda is sietünk. And yes, it can also mean that we do not only hurry here, but there too.
The only case where people might put is after a verb with a prefix instead of separating them is when a clause starts with ha or még ha:
- "(Még) ha elmegyünk is az orvoshoz..." -- "Even if we go to the doctor..."
But you can put is between the prefix and the verb here too, and, while I'm not sure about the rule here, I prefer it that way:
- "(Még) ha el is megyünk az orvoshoz..."
Yes, this "even if" kind of usage of "is" is kind of weird. I suspect that it is a contracted version of a more "regular" structure:
"Még akkor is, ha elmegyünk az orvoshoz...."
Very literally: "Even then, too, if we go to the doctor..."
And this gets shortened to either one of these:
"Még ha el is megyünk az orvoshoz..."
"Még ha elmegyünk is az orvoshoz..."
So, "akkor" (then) gets dropped, and "is" finds a new host.
The difference between "elmegyünk is" and "el is megyünk" is very very minimal at best.