The time gap can be explained because the Raspberry Pi does not have a realtime clock built-in. Raspbian takes effort to make the gap as small as possible and don't always start with 1970-01-01. It stores the date/time from the last shutdown and starts with it. So you will first see this date until the synchronization to the (internet) timeserver is finished....
If it's definitely not a cable issue, you can either install a remote desktop (if you are using ssh from another computer anyways) sudo apt-get remove xrdpand then use for example the standard windows remote desktop or use X11 forwarding how to use remote connections (stackexchange)
This answer is based on Buster, so Jessie may be a bit different.
Pcmanfm sets its wallpaper based on what the config file tells it to.
The config file is located at /home/pi/.config/pcmanfm/LXDE-pi/desktop-items-0.conf.
Simply change the wallpaper= line to the full path of the image you want.
Sounds like a ground loop side effect. I think that the Ethernet port on the 3B+ is not the isolated kind but just has common mode noise magnetics. So ground is DC connected through it and probably is making its way through a power supply in the Ethernet switch to AC power ground and thus to your HDMI monitor, then back to the Pi through the shield of the ...