Can the wire from the y terminal on thermostat be connected to the c terminal on heater board as pictures show?
I agree with @RetiredMasterElectrician that if the repairs were attempted by licensed HVAC contractors, they should have no problem diagnosing and fixing this and should do so free of charge since it will have taken them 3 tries to do so.

If the yellow wire at the control board was previously connected to Y, I have no idea why he would've removed it. That will be your call for cooling (turns on the compressor) and probably why your A/C stopped working (but fan/heat work which are controlled by G and W respectively). 

Yellow should be reattached to Y, leave brown connected to C, and everything should function. Do not jumper Rh to Rc. The Nest has an internal jumper when it is required. Plugging into the Rh terminal is appropriate in your case.



Before you had a C wire, the Nest will use the furnace's R/W wires to charge its internal battery. After you turned off the pilot light, it was probably only running on battery and couldn't maintain its charge. Using a C wire should fix that issue and give the Nest constant power.