Won't turn on

SOURCE: Screen goes black but computer is on

I would turn the computer off at night, but try to restart the computer. the computer can get stuck in a suspended state of "hibernation" and due to a program error or a corrupt file it can get stalled this way ( just like when your computer freezes)

SOURCE: when comp pluged in just a green flashing light

If you are sure that the 110 volt power is coming into the computer then the most likely problem is that th epower supply died. Obtain a new one of the same or larger wattagge as the one you are replacing . It does nto have to come from the manufacture of the computer. 

SOURCE: screen seems to be not working properly

try using a different monitor, borrow one if you have to, if the monitor works on your computer, then it simply means you need a new monitor. As far as the computer shutting down, it may be recording the bad monitor as an error in the loading process and automatically shutting down.

SOURCE: Gateway GM5045e Computer runs but no picture

Your video card may have gone bad but with out having it on the bench its hard to say at the moment.

SOURCE: My Sony Vaio PCV-RX860 won't boot and makes a clicking sound.

It's usual that new desktop PC's have fans sounds and keyboard LED's flashing during booting..There's nothing wrong on that, as long as you can get in to your PC OS.