Is there a mod that essentially clones the quick reply box?
How it would work is one quick reply box is static below all the messages on the page, whilst the other floats on the screen, disappearing once the static quick reply box is in the viewport. The floating quick reply box only appears if a user selects the "Quote" button on a message.
Additionally, how would you allow users to drag the floating quick reply box across the screen? I know the existing quick reply box lets you resize it, and that is easily configurable via CSS. This isn't required, just a desire.
The important part is that fact that the quick reply boxes are exactly the same, just cloned in the script. Both fields will have the same text in each and if you could display both at the same time, you would see the changes in real time too.
If you're only going to be displaying one reply box on the screen at a time, you don't need to clone it, just move the existing one around.
In any case, for all you're asking you'd have to get it done using javascript.