Bearing cone and locknut won’t tighten. Easily fall off when taking wheel off. Has it snapped?
Yep, axle is broken.

Do not ride the bike with the broken axle! The axle is supposed to support the frame on the wheel, but you are now asking the quick release skewer, which has much less material and strength, to do that. If the skewer fails, the wheel will come straight out of the dropouts and a nasty crash will likely ensue.