I agree the panelboard is rated 225 but the breaker is sized at 150 so 3/0 is plenty good....
The neutral is stated on the plan so you must use that size however there is a 100 amp breaker with 3/0 wire and a #8 equipment grounding conductor. I think that #8 is wrong as it must be upsized since the feeder size is larger than needed 250.122(B)