Or there a way to send SMSs through a PC programatically? Alternatively: is there an android application that exposes an API over a socket (REST or otherwise) that allows one to send SMSs (and possibly other functionality)?
Again, I can suggest GTalkSMS which is like a remote control for your phone over XMPP.
Disclaimer: I am involved in GTalkSMS
CHeck out this one.
Zipwhip Text Messaging: LINK.
This application supports user to send text from PC over a HTTP and client app. Also, it supports PC and Mac.