Can SQL Server (2012-2014) be configured to run "on-demand?" I am very, very new to SQL, and am having difficulty configuring the server appropriately. All I have to work with for now is a small copy of the database schema of the company I work for, with randomized data in it. Running a small server with just a few tables that I only occasionally query to test applications, can I configure SQL to use resources "on-demand" (perhaps an agent that listens for queries, and does only this, not caching data in memory) or to use minimal system resources? 

I find that, even with only 15 tables totaling 250MB in storage, SQL is always using at least 2GB of memory on a machine with only 8GB to spare.

I did look for similar questions here on StackExchange, but only found debates about how much memory to devote to the OS as opposed to the database on a server machine.
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