Is it possible to post a question to Stack Overflow using PHP in the same way that it's possible to post content to Facebook?
As of this writing https://api.stackexchange.com does only allow the writing of Comments.
Starting with version 2.1, the Stack Exchange API provides limited support for writing to sites in the Stack Exchange network. There are a number of constraints imposed to prevent abuse and loss of quality on Stack Exchange sites.
And also http://blog.stackoverflow.com/2012/09/stack-exchange-api-v2-1/
Oh, and our first set of write methods.
It’s now possible to create, edit, and delete comments via the Stack Exchange API We’re starting small, with the least important of our content, to safeguard the quality of our content.
4Technically it supports more, since it's what the Android app alpha uses behind the scenes, but...shh, secrets.– Tim Stone ♦Aug 13, 2013 at 19:32