"Animated backgroud is very distracting. I'm constantly reacting to the new icons floating into the screen."
Another user pointed this out and a toggle for the animations will be added to the settings so that you can turn them off.
'Underlined "Or" in "Click Here Or Drag & Drop To Start Uploading" makes me think it's some kind of a link. Any reason to underline it?'
It's underlined just to separate the two options (clicking and dragging) for those who just read the "Click Here" part and assume that the rest of the sentence is just a description to why they should click here (e.g "Click here to start uploading your file").
"If I upload multiple files (which worked well) I want to be able to copy all the URLs at once. Displaying them in a text box would be good."
Instead of a text box there could be checkboxes next to each file allowing you to copy URLs and delete files in bulk. We'll work on adding this as soon as possible, thanks for your suggestion.