--budgie desktop settings can hide icons.
--to delete desktop items, delete them from the nautilus (file manager) desktop folder, not the desktop.
I miss desktop click-and-drag, too. A lot. It was recently done away with (do a search here for reasons why) and I don't believe it's coming back. I really miss having my garbage can on the right edge, centered, but other that that it can be lived without. You get used to operating out of Nautilus desktop folder, though.
edit: added an omitted word