I tried connecting a friends dualshock 4 to my PC w/o a dongle, and was having some issue. The stuff you explained to me feels minor, my steam is always open and I wont need to play unplugged for more than a couple hours.
I do not believe that gamecube adapter will work for pc. There are adapters for pc, but I know the official ones for the WIIU might get grumpy. If you have it, give it a shot. Why spend money when you can try something else you already have first?
Dualshock should connect through bluetooth as well. The ps3 controllers have shoddy connections with it though the ps4 controllers should be fine.
i plugged the bluetooth dongle in the back of my pc and i had some connection issues.
i plugged it in the usb on the back of my keyboard and i had no problems since then.