LTC3534 - Buck-Boost not working
I have prior tested the board without the output capacitor C22 mounted
  on the board. (I forgot to solder it) Could have this damaged the IC?


Yes, this could damage the device because the inductor current/energy has nowhere to "dump" and it will generate a fairly sizable peak voltage. Difficult to be exact.

With no load, the device MUST operate in burst mode control, You have PWM mode set to prevent burst mode and this is a likely problem. If load currents are less than 1 mA you must run in burst mode - see the top 2 graphs on page 4 of the data sheet.

However, I think the main issue is running it without the output capacitor.