Don't be avoiding freeweights so much, mate. Not saying machines are bad, far from it in fact as sometimes machine exercises can help work a target muscle better than their freeweight versions. Sometimes the opposite is true though, e.g. dumbbell flyes are always going to be a better chest builder than a flye machine. Surely there's gotta be a bench free at some point?
I'm definitely not a fan of working biceps before rows - the biceps are often a weak link in rows as it is, best not worsen that by fatiguing them.
There are a ton of better exercises for the shoulders than upright rows. Take your pick.