Is there a public API server for Stellar? I'm writing a Stellar application. I'm looking to use the API provided by stellard without running a copy of it myself.

Is there a public API server for Stellar?
Yep:
https://www.stellar.org/api/

Stellar Foundation runs two instances of stellard you can use if you
don't want to run your own.
network   address           RPC Port  Websocket Port
test          test.stellar.org 9002          9001
live           live.stellar.org 9002          9001