What could be causing GFCI breakers and receptacles to trip?
Sounds like the circuit is sharing a neutral with another circuit. 

With your GFCI breaker installed, turn off all other breakers. Reset the GFCI breaker, which should now hold. If it does not hold, then you have a problem with the wiring on that circuit and need to open each box to investigate. 

If the GFCI breaker does hold, turn each breaker on one at a time until you find the circuit that trips the GFCI. This is the circuit with a shared neutral. 

You then need to open switch boxes, light boxes, on both circuits to figure where the neutrals were wrongly tied together. Suspect anything that looks like a homeowner addition, or fix. A common place this might happen would be in a switch box that controls different lights, with two or more circuits feeding the box. An unknowing DIY-er may have tied all the neutrals together.