Still haven't told me your iMac's screen size?
Does the second monitor show any of the desktop?
Here is the procedure for getting the second monitor to work.
http://support.apple.com/kb/HT5019
If the second monitor still isn't working, it maybe that the monitor is too big and the video card cannot drive a second monitor that large.
On my 2009 iMac, for example, with the 256 VRAM card I have installed, I can only run a additonal monitor that is within 20-24 inches in size with a resolution less than what is on my main iMac screen.
If this is a new iMac and you knew that you might want to run a monitor of this size Apple had a build to order GPU option of a 1 or 2 GB VRAM graphics engine, but I think this might only be available on the 27 inch models only.
As far as RAM goes, you do not have nearly enough RAM to run both OS X and other applications efficiently and smoothly.
OS X, by itself, needs a real world working amount of 3-4 GBs of RAM to do it stuff. It simply isn't enough for running the OS and any other applications.
Your iMac can take up to 32 GBs of RAM.
I recommend, at least, installing 8 GBs of RAM and maybe consider installing 12 or 16 GBs of RAM.
If you are using any apps that are really CPU, GPU or RAm intensive 8 to 12 GBs of RAM should be fine.
RAM is pretty cheap for these machines.
You can get reliable RAM from online sources Crucial memory or OWC.
Both sites have a method for helping you determine the specs of the RAM you need to use in your iMac.