I used to do shake the wine thief up and down to gather enough wort. If there is any oxygen uptake, I’ve never noticed any oxidation from doing it. I now have a conical fermenter with a ball valve at the bottom that I pull the sample from. One could argue that I could get oxygen pick-up from that as well, but again, I’ve never noticed any oxidation off flavor or other ill effects from doing this.
I guess it depends on how vigorously you’re shaking, but I stick my hydrometer inside the wine thief to take a measurement. Fill first, then drop the hydrometer in - there’s always enough to float it though that would depend on your fermenter.