Coffee based beverages don't explain childhood and teen obesity. Those two categories don't drink nearly as much coffee as adults.
Yes and no; if you go into Pret with a kid they will give you a free "babycino" which is some sort of caffeine-free but presumably fat- and sugar-laden hot drink. Kids are being trained early on to drink this stuff.
Besides that, some categories of obesity have plunged:
"U.S. Childhood Obesity Rates Fall 40% in Decade"
http://www.wsj.com/articles/SB100014240527023048347045794053...
Adult obesity rates have leveled off for the last several years. I don't believe it's a coincidence things are moving in a better direction at exactly the same time Americans started drinking less soda (such that both Coke and Pepsi have been seeing declining sales in the US market for the last few years). What Americans aren't doing less of, is consuming fewer coffee products - they're drinking vastly more than they were ten years ago. If that were fueling obesity we'd be seeing a continued increase in obesity among adults.
Further, the obesity rates skyrocketed starting from the early 1980s. They trace almost perfectly inline with soda, fast food, and high fructose corn syrup. They don't follow any line with the vast array of coffee products, which only took off in the mid to late 1990s.