Sounds strange. Could be that your software is not working very well. You could try a reset of all settings or restoring the unit as new to test. Furthermore check the power button is not sticky or held down slightly. If it is use an alcoholic wipe to loosen it up.
If your machine is a retina MacBook there is no hard drive cable. It's an SSD that slots into the logic board. Likelihood is that the disk needs repartitioning and a new OS installing. In some cases Apple use "flash storage repair" diagnostics at the Genius Bar. Maybe worth checking your one does/doesn't need it
Sounds like potential garbled video/distortion. Could try a 2 button reset and see if it improves. From working on the Genius Bar, it's progressive and results in a unit replacement. So could need a logic board replacement worst case
No light from MagSafe suggests the board is dead, however you could try... Use your original MagSafe board or the new one and short the connection on the logic board using the Power On Pads. This will tell you whether your top case or logic board needs replacing. The power on pads are usually located near the top case connector. Use a T8 screwdriver or other that is big enough to cover both the pads. Plug into power and have 1 fan connected. Wiggle the screwdriver over the pads and hopefully you get a spin from the fan.