I signed up for a US Airways, now AA Aviator Credit card and earned 50,000 miles. When I go online to redeem there are two choices of seats, Economy Miles Saver (none available for my dates) and Economy AA Anytime (some available). Am I eligible for both of these?
These are just different tiers of rewards. Like most frequent flyer programs, AA has a tiered system where the Saver awards (the ones that you rightly note are not available) are cheaper, but go faster. Look at AA's reward chart for the most up to date info.
American Airlines draws no distinction between miles earned from credit card bonuses, credit card spending, or any other partners (for the purposes of elite status asssignment it is a different story). An AAdvantage redeemable mile is an AAdvantage redeemable mile. Put another way, the miles you earn from the card go into the same co-mingled pot as the miles from your other activity, whether from flying, bonuses, partners, customer service credit, transfers, or gifts.
The AAdvantage Terms & Conditions do not indicate any restrictions on who may redeem for the general award ticket types. MileSAAver awards are somewhat analogous to discount tickets; they are tightly capacity-controlled and heavily restricted. AAnytime awards are somewhat analogous to full-fare tickets; they are much more costly (generally double the redemption level by the award chart / redemption table), but are flexible, can be used on any available seat, and come with a few perqs like access to "Preferred" seats in the economy cabin.
In fact, holders of AAdvantage credit cards, including the Barclays Aviator cards with annual fees, may qualify for additional discounts for MileSAAver seats to certain destinations under the Reduced Mileage Awards program.