You always, always, always check in with the operating carrier (or the carrier whose name is painted on the side of the plane in the case of American Eagle/Alaska operated by Horizon or Skywest/Delta Connection/etc) of the first segment on that day/set of connecting flights. This applies both to checkin at the airport and online checkin. Therefore, you'll check in (online and/or at the airport) with AA for your MEI departure and with AS for your FAI departure.

Both airlines will be able to check bags through to your final destination (as is essentially always the case whenever you buy a single ticket issued by any airline -- exceptions to this are very rare); both airlines will be able to issue paper boarding passes for all flights at the airport. I don't think you'll be able to get electronic boarding passes operated by an airline other than the one you check in with, and you typically can't check in with the operating carrier for segments that are operated by a different carrier than the one you start with. That is, you probably won't be able to get boarding passes from online checkin for the AS-operated flights on the outbound or for the AA-operated flights on the return.