First of all, thank you for taking the time to read this post. I'm wondering would I be able to run Battlefield 3 with the specs below and if so, what settings do you think would enable the smoothest game play?
Windows 7 Home Premium - 64 bit
Intel(R) Core(TM) i5-2320 CPU @ 3Ghz (4 CPU's)
4096MB of RAM
NVIDIA GeForce GT 555M graphics card
Current display is 1920x1080
Also, just a question regarding my processor. Is it a Quad core?
Yep, your CPU is a Quad Core. (Quad = 4, and you have 4 cores)
Generally, I would not advise playing a game like BF3 on a laptop, unless its very high end. Laptops are unfortunately inferior to desktops even if they have the same specs, for example:
A Desktop GTX 550 > A laptop GTX 550m
My friend recently bought a Dell XPS 17, with an i7, 6GB RAM and a GT555m and a 1920x1080 screen - so similar to yours, but his performance on less demanding games such as Call of Duty was not all that great.
However, you should still able to run it at a playable standard with mostly low and medium settings.
As Nicklovn has said, playing it into a 720p TV might be an idea, as your screen res is very high for a laptop and will therefore be quite demanding on your hardware.