RtCtrl RtAlt RtWin Fn two diodes are sideways and one end is not attached to the pad. There is also a missing diode in this area.
There is a pile of solder to the left of the right control that should be removed. I suspect you need a diode there.
Fn key is bad because of the sideways diode
Enter key - one end of the diode either doesn't contact the pad at all or has poor contact
7 8 0 - = Backspace sideways diode, check upper poles of diodes,
7 8 check upper pole of the diode, switches need re-flowing, solder joints are probably cold
U I O P appears that the upper pole of the diode isn't soldered
I think several of the diodes aren't soldered on both ends. Multiple switches either have piles of solder or they open areas where it didn't flow properly ( like F D R T L alt L win )
missing the diode for L shift
there is more I could say but that will get you going.
I would take as much solder off of the switches as you are comfortable with.
Next reflow every solder joint on the switches. You do realize that none of the switches are seated properly?
Then get a loupe
(
http://smile.amazon.com/Illuminated-Magnifier-Construction-Kare-Kind/dp/B00YBHQ7X2/ref=sr_1_3?ie=UTF8&qid=1464218659&sr=8-3&keywords=jewelers+loupe)
and look at all of those diodes. You are missing two I believe, two or maybe three or sideways, and several don't appear to have been soldered on both ends.
You also have two or more melted stabilizers.
If those are Zealio's- given the clear bottom - you probably have some ruined switches. If they get too hot (obviously some did since there is melted solder on them ) they bottom of the switch can warp distorting the contacts. They will either "chatter" or just not work. I will bet you that is a problem with some of the switches.
Don't worry about a multimeter yet, just fix the basics and see what you have. If I'm right about these being Zealio's, pull every switch that has solder on the base thatdoesn't work after you done the preceding steps.
I hope you didn't pay too much for this board - I would probably throw it away. I could fix it but it would be a real pain in the ass.