There is nothing illegal about getting a card for the bonus, then cancelling and reapplying. However (a) most issuers have a minimum period between applications for the same card to get a bonus and (b) some (not many) might actually take back miles if you cancel your card too quickly. If you keep it 6 months you will almost always be safe from a clawback.

You might think of the value of a mile as something between 1 and 2 cents - depending on your travel patterns.

Concentrating your travel on one carrier has the advantage of gaining elite status, which has many advantages. One of these is usually bonus miles, i.e. a 25%, 50%, or 100% bonus on miles flown. If you spread your travel over different airlines, you may sacrifice not only the miles gained on your primary airline, but bonus miles for achieving elite status, or a higher elite level.

Serious "Road Warriors" might fly 150K miles a year, have a high elite status and get a 100% bonus, perhaps sometimes an extra bonus for buying high-fare tickets, and get 300K a year or more points from flying. Most people who don't do that get the bulk of their points for signup bonuses.

I would suggest getting as many credit cards as your credit score and ability to meet the minimum spend will bear. If you choose to fly AA mostly, there's no reason not to get a United card, which gets 55K points if you do it right, and then you're within spitting distance of an international ticket. A USAir credit card gives 40K, plus 10K each year, so after having the card for just over 2 years and paying $178 in fees you again have enough miles for an international ticket.