Ok - the green thing is a capacitor, its there to smooth off any signals, youll get 0.2v or so on one side as this side goes to ground.

You need to test the chip itself and yes - can be done with a DMM. A logic probe makes this kind of testing much easier - well worth investing in.

On DC volts - lowest setting. Put the black lead on pin 8 - ground and then go round the chip with the red lead, pin by pin.
IC15;
Input pin 3 should be high, output pin 2 -low
7 high, 6 low
14 high, 15 low
9 high, 10 low

IC16;

3 high, 2 low
7 high, 6 low
14 high, 15 low
9 high, 10 low.

These are the switch inputs - make sure these work OK.

The switch outputs are IC17 & 18 - these are 7406, slightly different configuration, pin 7 is ground.
IC17
Pin 1 high, 2 low
5H, 6L
13H, 12L
9H, 8L

IC18

 same as IC17.

Let me know whjat you find, from this we can see if buffer chips are ok and if PIA is working OK