I just received a cutting with an Inflo. Should I remove the whole stalk? Or just the branches of the stalk leading to the flowers?

I have done both. Cut all leaves and leave inflo and cut inflo. The new roots can support an inflo but will not grow leaves, or cut inflo and it will grow leaves. Either way if it takes it will survive. If you leave inflo it will probably not flower next year. I've had luck both ways. I think the proper way is to cut inflo and most leaves but I grow for flowers so it's hard for me to cut an inflo.😎
Good luck which ever way you go.

Do I cut the whole info stalk down?

Yes, that's what I do, how far along is it.

I would only leave about a half inch nub. That will dry up and fall off on its own.