1. Very very slightly - you produce more saliva which you swallow (more on this in 2), but I'm not sure if that really contributes on a meaningful scale.
2. I'm only talking about the important and/or long-term effects:
-Vastly better dental health (depending on what your gum contains, xylitol and polytol are good components to have)
-Chewing gum might exacerbate a problem for a person with some sort of Temporomandibular Joint Disorder (TMD, TMJ)
-I assume the gum you're talking about is Wrigley's Doublemint - like many gums, it contains Aspartame, which is a controversial artificial sweetener. Many industry funded studies show this substance is safe, but over 90% independent studies show adverse health effects. There are some other sweeteners too, but for the most part they don't have adverse effects, especially in chewing gum doses. Some gums just use plain sugar.
-Along with the production of saliva, having something in your mouth triggers the cephalic phase - gastric, pancreatic and biliary fluids being released in your body in preparation for digestion. There's a minority of people claiming this can upset their stomach (changing the acidity), but I can't find any studies backing this statement up, and it seems like this happens to a very small amount of people anyway. If you eat the gum after lunch, I imagine this can only help with digestion though.
The reason we get drowsy after a lunch is because the body draws blood away from the brain to help the digestion process.
-Eating food that is easy / quick to digest will reduce this drowsiness. This also applies to amount of food eaten. In my personal experience eating light food (I buy chicken salad or an italian wrap!), and eating enough so you are just below the feeling of being 'full' helps tremendously. Try playing around with this and see if it works for you too.
-A 5 minute very light exercise session that will get your blood running. I use a local pull-up bar for a set or two when I have the time. This is just to 'warm up' and shouldn't be actual exercise.
-Splash cold water on your face for temporary mental sharpness.
-Stay hydrated, drink a lot of water. There are almost infinite reasons to stay as well hydrated as possible continuously. One of the first signs of dehydration is fatigue.
Hope this helps you!
Sources and detailed info:
Polyol containing chewing gum helps prevent dental caries:
Xylitol containing chewing gum grants various boosts to dental health:
Contents of Wrigley's chewing gum (look up the ingredients for your gum of choice): http://www.wrigley.com/global/about-us/ingredients.aspx