I can help you with number one, number two I'd have to see more angles and photos of both sides of the mating connector. (EDIT: There's magnets and a weight that makes those things retract.. is the weight caught up on something under the sink?)
I had a handle that did this too. This is caused by one or many occasions when excessive force is applied to the handle, breaking or bending the metal inside.
There's an set screw hidden right below the red/blue dot, use a razor to catch the edge of it and pop it off. Watch your sink drains, they're like magnets for that darn dot. Once you back out the set screw about 3/4ths of the way, the handle will come right off the top.
You'll find the inside of the handle has a hollow rectangle that slides over a solid metal rectangle on the valve. You'll likely find that the metal of the hollow rectangle has been bent.
I fixed mine by using a vice grip to squeeze in and tighten the sides of the female side of the rectangle back up. Test it for fit, make sure it's good, then take it back off and wrap it with steel wire and epoxy. I must have layered mine in about a quarter inch thick. Make sure to leave clearance for the set screw, and definitely leave the screw in place to protect the threads from the epoxy in case it drips.
The idea is to reset and strengthen the metal in place. Wire alone won't do the trick, since it'll round right out too, the epoxy gives it the strength. You might even be able to use stranded copper wire and solder to replace the epoxy. There's more than one way to skin a
Alternatively, a lot of faucets have lifetime warranties that they hope you don't remember about. I once called up the manufacturer and got a brand new handle shipped out.