The preposition "bei" always has a notion of "geographically close to". So "bei der Universität" would mean "right next to it", but not inside, or at another institution nearby. You can however say: "Ich studiere bei Professor X" if you write your thesis at their chair, as in this case you will be close to that profesor quite often.
When talking about institutions such as schools, universities and such, you use "an" to say that you are working/studying at this institution. But if you are just talking about the building of that institution, you use "in". A university usually has several buildings, so "in" only works if you specify the building.
So my translations of your sentences are:
I am going now to Stanford University. --> Ich gehe jetzt zur Universität Stanford.
We are studying German now at Stanford University. --> Wir studieren/lernen jetzt Deutsch an der Universität Stanford. (depending on whether you major in German or just attend a language course)
We are eating noodles at Stanford University. --> Wir essen Nudeln in der Mensa / auf dem Campus der Universität Stanford. (Note that I inserted "in der Mensa" or "auf dem Campus", because a university as such is not a place to eat. Technically, you could also say: "an der Universität", but that sounds a bit weird to me.)
Are you studying French at Stanford University? Studierst/Lernst du Französisch an der Universität Stanford?