*A always works on Operating Carrier, not Ticketing Carrier, so the CO earning chat will apply for your CO leg.

H is an economy fare class. If you have a revenue first class ticket it should be one of F, C, J, D, A or Z so I suspect that the agent is right about it being booked in A class.

Unfortunately this list only changed to add A in Feb this year on CO so Agean may not have made an update to reflect that A is now a first class fare code. If so there's a risk you will get no miles at all for this segment.