You shouldn't be worried because the puffy lip may have been caused by aggressive kissing, which is a common occurrence. It may have also been caused by an allergic reaction, which is easily treatable. You should see a doctor immediately, so as to find out the cause of the swelling and how to treat it. Check out http://www.buzzle.com/articles/swollen-lower-lip-causes.html for details.
I agree with the other answers. Abreva and time. Should be ok. As for catching an STD from heavy kissing alone, it is definitely possible if the person with the disease has inner mouth sores, bleeding gums, or outer active sores. Frankly, if you're worried, wait a week or so and get the standard tests done for a variety of STDs. (for several it's a simple blood test) --- ANYBODY who is active should be getting tested regularly. Make it a habit, every few months. Most STDs are quite treatable -- but you have to know that you've caught one! Contact the CDC (Center for Disease Control) for a clinic near you. Some do it free or really really cheap. Planned Parenthood also does testing, always confidential, under 18s tested without parental permission in many (but not all) locations.