1 & 2: The photo is a bit high angle to see whether anything is broken, but that switch definitely needs to be clicked by the screw just as the extruder nozzle tip touches the bed. If you need to add something (like one of those brass standoffs from the bits left over from building a computer) to make that happen, even just for the moment, then please do so. I think we could confirm if you share a picture of a nice close front view of that switch and screw with the bed at the top (where it's jamming!)...

3. The alignment between the z thread and the nut will be the worst at the bottom of the travel. If the bed has hit the bottom of the case, it may be skewing the bushings on the rail which could contribute. Also, the lubrication is probably not so fresh down there. It's likely just stuck - it's making the buzzing noise meaning it's trying to move, but evidently having trouble. With the printer off, manually rotate the z thread (clockwise, looking from above) to raise the bed a bit, and then power up and try raising the bed (down arrow in RH remember ) again. You should be able to fix this in future, but save that for when you've got this z limit switch issue sorted!

1. Home Z
    After clicking the Home Z,
    The printing bed still keeps crushing the extruder, due to limit switch is not sensed.

2. Z Limit Switch
    Even when I screw it low enough, but it cannot reach the limit sense.

    i.imgur.com/R40LumJ.jpg       (Is this screw too short ?)

3. Printing Bed Never Goes Up (New Problem)
    After my maximizing Z distance from the nozzle,
    This time, the printing bed becomes no response and never goes back. (totally losing control)

    youtu.be/N_Ayl0Tyc_0

    The newly arrived Solidoodle 4 is killing me...

2. Yup as wire10 suggests, look for the little switch on the left hand side at the back of the bed. There's an adjustable stop in the back of the case that this switch should run into when the bed rises - if this is not aligned correctly and the bed is hitting the extruder before the switch contacts, it will make some horrible noises. Don't worry, you're unlikely to have damaged anything, it just needs adjustment. Post a pic of that switch arrangement when you find it, we'll tell you if anything's really wrong, otherwise just wind it down and look up 'z-tab adjustment' on the forums.

youtu.be/niXkCpztmj0

As the video shows...
When I officially print objects, the printing bed always moves toward the sky and never stopped.
(Even after my adjusting the Z offset as the attached photo, the problem remains the same.)

1. Is it right ?
... when I use Z Manaul Control in Repetier to go down but the printing bed goes up?

2. Why the Printing Bed goes up without Control?
   ... I didn't change any setting