The next step secret to getting more of this faucet apart are the two slots cut across the cupped part. See the arrows in the following picture.
It looks to me as that the cupped part is threaded down inside the larger diameter part that sticks up from the hole in the sink. There is probably a special tool used to engage these slots to thread the part out. It may be possible to lay a piece of metal bar stock across these slots as a tool. If you do that make sure that it has nice square corners.
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 cat faucet.
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.
Best Answer
YES and YES
YES and NO - you should go ahead and try tightening it up with the nut, it should work. Those types of faucets normally have a gasket under the upper portion and no plumbers putty is required.
NOTE- it will likely be difficult to tighten that nut, most of those types of faucets come with a manufacturer-supplied special tool to reach up and tighten the nut with. If you don't have that tool then you will need to improvise.