The Starbucks at DTD is like any other Starbucks--you can use your Starbucks GC, but not Disney GC. You'll earn stars for your purchase and can use/redeem your free food/drink rewards.
The Starbucks at DCA and DL accept both DL and Starbucks GCs. You can earn stars if you use your Starbucks card, but cannot redeem any rewards. Also, you will not earn any bonus stars if you have any special offers on your card. For example, get 3 bonus stars if you purchase a breakfast sandwich--you will only earn one star, not the bonus.