Can SQL Server (2012-2014) be configured to run "on-demand?"
The best fit for your requirements is probably SQL Server Express LocalDB (2012 and later).

It has a very small footprint, does not create a service, starts on demand, aggressively releases resources, and automatically shuts down a few minutes after the last connection to it is closed. LocalDB is specifically targeted to developers testing on a local machine. Things like external connectivity and SQL Agent are not available.

More information:

Introducing LocalDB, an improved SQL Express
SQL Express v LocalDB v SQL Compact Edition
Getting Started with SQL Server 2012 Express LocalDB