> I see no practical difference between the paper and the styrofoam

Styrofoam provides much more insulation.

I'm not sure if that's an issue. If you get a Tim Horton's tea in a paper cup, it's still scalding hot even after 10 mins (and it tastes like piss, so I wouldn't recommend it BTW). If you want your drink to be hot 2 hours later I'm not sure a styrofoam cup is going to do much better. You probably want to get them to put your drink into a travel mug.

No, insulation between your hand and the hot drink so you can hold it comfortably.

Coffee places usually give you cardboard sleeves where your palm goes, that help with insulation.

If your hand is big enough and the cup is light enough you can support bottom of it with your pinky and the top of it with your thumb while horizontally balancing it with the rest of your fingers, you have three-finger redundancy if your coffee is too hot. The problem with the latter approach that little finger supports most of the weight of the cup and I can't hold more than a medium coffee.

The keyword here is practical :) In the time it takes me to drink my Jamba Juice, the melting aspect is impossible to differentiate.