It's the airlines that are the members of the OW alliance. You have mileage accounts in AA (and QF?)

With enough miles/points in either account you can obtain award tickets for domestic or foreign travel. You must follow the rules and use the miles within each FF account; they cannot be commingled for award travel. If your award travel is to destinations served by OW member airlines other than the one whose miles you're using, that can be arranged. Look at the earning/spending guidelines in the respective mileage program(me) rules on the airlines' websites. There are also specific oneworld mileage based awards in which the amount of miles traveled governs the miles consumed from your account. These award rules are also listed (or linked) from the AA or QF mileage plan websites.

If your flying is as infrequent at you say, then obtaining or retaining elite status with any FF plan is going to be difficult. AA makes allowances for "challenges" to higher status (see the stickey threads on the American board) but if you do not follow up the challenge with a fair amount of travel in subsequent months/year, then you will lose the status sooner or later.

Also, if your long-haul travel occurs at the end/beginning of the year, that's an issue with some plans that reset the "clock" for elite status at year end.

You should do some research on both ther AA and QF boards here at Flyer Talk to get familiar with the lingo and rules - it can be quite rewarding. And lastly, welcome to FT if you haven't been welcomed already.