Mono is a very contagious disease and something like 80% of the people in the world are carriers. Being a carrier means that you have been infected at some point in your life. Of the people that are infected, only about 50% of them ever develop symptoms. That means half the people infected with it will actually come down with symptoms and be sick, while half the people infected never get sick and usually never know they are even infected. Mono is contagious in the first 30 days after someone has been infected - the tricky part is it usually takes 30 days after infection before you develop symptoms. So you are contagious before you even realize you are sick.
As to your question about being tested, I believe they can tell you the same day whether you actually have mono or not. While your symptoms do sound like they could be mono, it could also be anything from strep throat to allergies. Regardless though, if you think you have been exposed to mono, be sure to mention it to your doctor so that he/she will order the blood work.
As far as I know, mono is not a seasonal disease that once you are infected flares up and makes you more likely to infect someone else with. So as far as your girlfriend transmitting it around spring - I don't really think that is a true statement. Once you are infected with mono, you can not be reinfected with it. It stays in your body the rest of your life, therefore you are never "cured" and are always infected.
Good luck with the doctor - hope you start feeling better soon