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@JenniferW You have to decide what your risk tolerance is. The general rule I follow is to segregate the cards I use for my regular use from the cards I use for piggybacking. But you don't strictly have to follow that guideline, and there can be situations where it makes sense.
Losing cards happens, but it's not particularly common in my experience. People post about it on this thread because it's newsworthy, but people don't post when their cards don't get shut down. Also, I haven't seen many data points, but it does seem that at least some of the time you can get another card with the same issuer shortly afterward.
It boils down to risk / reward really.
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I think New Company stopped using AMEX a bit ago. AMEX cards generally weren't posting very consistently IIRC, which is just bad for everyone. I don't know about Old Company's current rules.
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To answer your question, when you downgrade, you don't lose any AA miles that you've already earned. Typically when downgrading, the benefits like mileage accrual or cash back on the no fee card isn't as good as the $95 or $99 a year version; I don't know offhand if that is true specifically for the Barclays AA card but it's a general pattern I've seen.
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I don't usually worry too much about inquiries. For me they only drop my score by a few points each, and I've gotten cards approved with more inquiries on my report than you have now. But it does sort of depend on your overall credit history, and so your experience will probably be different.
One thing that is probably more important is to know what each issuer's rules and limitations are. For example Chase has a famous 5/24 rule, and other issuers have other limitations on how many cards you can apply for within certain timeframes. There are lots of websites out there that discuss this sort of stuff; personally I like Doctor of Credit.