Microwallet in my opinion has the best/modern one, if you do not play with the layout too much, the ad placement is good for converting into clicks.
https://www.microwallet.org/api
Does the Microwallet faucet allow for multiple coin types? For example, if I wanted to setup a DogeCoin faucet at doge.mysite.com and a MintCoin faucet at mint.mysite.com, could I do it (with two separate databases and installations, of course)?
Also, can users use their own wallet addresses?
Microwallet accepts normal Bitcoin addresses, yes. But it doesn't support alt coins. MasterFaucet has built in support for some alt coins, and I think MyFaucet could work too if you modify it.