Whow, that's really great!
By looking at your pics of the tangled drill I thought immediately: you need a cap for the tip of the drill. Instead of just placing that cardboard disk at the end of your spool-to-be, how about using one of that cardboard tubes (part of) that's big enough to fit over the end? Close one end with that disk you already have and put it over there, open end first, so it covers anything between your end-of-spool and the body of the drill. No gaps for thread to get tangled into. And if it tangles on the cardboardtube, just brush it off.
Speaking of, you even could make some tiny little cardboardtubes just fitting your dowel. You fit them onto the dowel in the drill, spool your thread, get your regular spool, fit a wheel on one end, then match it to the drill and gently push the cardboardtube with the yarn onto the final spool, put a second wheel onto the open end - spool done, tool ready to go for more.
And for the toilet paper stand - you just need something HEAVY on its foot to keep it down. Or use the handle of a door to put your big-cored-yarn onto - they are not likely to give in so easily.