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Yes, for codeshares or not codeshares, same advice always applies, globally: you check in and do clearance with whatever metal you are about to clamber on. So at DUS and LHR this is a BA operator, at JFK you go over to AA.
For VeriFLY etc, you need the BA PNR, not the AA PNR. Hopefully you have that, AA can communicate this via Twitter if necessary. You should be able to use VeriFLY in the PNR mode to enter in your details for the London-New York service, which is the one that matters. Because of this mixed ticket I wouldn't worry about the onward flight, you can sort that out in LHR. There is a high chance that OLCI won't work, so as you indicate, go to BA at DUS and I'm sure it will take very little time to sort out your flights.