Unfortunately, the baggage thing is a deal-breaker for me. In other
words, if I don't know ahead of time what the exact rules are, I don't
feel comfortable booking with that airline.
What I'd prefer: a free checked bag, a carryon cabin bag and a
personal item. And a seat I can choose without having to pay for it.
Sandra, if you're concerned about the specific carry on rules like carry on measurements and weight, the rules of the actual carrier apply so in your case, British Airways' rules apply, not American Airlines. When you're at the gate, you might see a baggage template of BA's specifications. The gate agent might have you put that bag inside the template to prove it meets BA's rules. If the bag doesn't fit, you'll pay BA's penalty, not AA's penalty.
For checked bags, when you buy your tickets on AA's website for a BA codeshare, the checked bag fee on the AA website applies. If there is a free checked bag allowance when you buy your code-share ticket, it will be accepted by BA.
As for not paying for seat selection, when you buy your ticket on the AA website for the BA codeshare, you'll see what the options and costs are including when or if there is a time perhaps within 24 hours of the flight that seat selection is free. Keep in mind, waiting that long could mean that whatever seats are left to select are scattered middle seats. That is always the risk of not paying up at the time of buying the ticket this day and age.