1. Obesity is increasing in the population, so it can't be genetic. Well, average height is also increasing in the population. Does that mean that you could be as tall as me, if you weren't too lazy to grow? Twin studies and adoptive studies show that the overwhelming determinant of your weight is not your willpower; it's your genes. The heritability of weight is between .75 and .85. The heritability of height is between .9 and .95. And the older you are, the more heritable weight is.
I think the analogy between obesity and height is weakened by likely differences in the effect on the variance of the traits due to environmental changes. First, remember that the term "genetic" is very broad, while the term "heritable" is very specific. Heritability is the proportion of trait variance within the population explainable by variance of genes. The said traits are usually thought of as quantitative traits, like height, weight or IQ, which exhibit a normal distribution. To say that a trait is .95 heritable does not mean that it is caused 95% by genes, that's not even wrong. Rather, it is to say that 95% of the variance within the population can be accounted for by the variance of genes within the population. But heritable traits are also usually affected by environment; if you starve someone they will be short, but retain five fingers. The number of fingers you have on your hand is not heritable, because there's no real variance within the population of the trait. It's genetically specified, but not heritable. In regards to height and weight, there are some specific details of difference that are important. It does seem that in actuality the average height of American whites has stopped increasing, and are about what they were a century ago (Northern Europeans keep getting taller). Beyond a certain level of nutritional input you simply aren't going to get any taller. (there are some discussions as to whether high levels of milk consumption are responsible for Dutch height, so what as opposed to how much may matter) Environmental inputs may have reached the era of diminishing returns on height in the developed world. The reason that the heritability of height is lower in poorer nations than in wealthy nations (where it is almost 1) is that in the former there is still environmental factors which account for some variance in height. Some people in less developed nations don't get enough food to realize their full genetic potential. Naturally the same applies to body weight, heritability should increase as there is an equalization of food intake between the poor and the rich, so that more of the remaining variance will be accounted for by genes because the genetic variance remains the same even when the environmental component is removed from the equation. Though I suspect we've come close to the point of zero diminishing returns on the population wide environmental inputs which might affect height in the developed world, we're not there when it comes to mean population weight (or BMI). After all, people can take weight gainers or go on diets and exhibit a lot of temporal variation in phenotype in a way that does not occur with height after adulthood (aside from the small but non-trivial gradual decrease in height due to spinal compression). While I assume that the environmental parameter of most importance for height is food, weight (or muscle mass) can be changed through both physical activity and the quantity and nature of food. So if there is such potential plasticity in weight why is it that the heritability is so high? The important point to consider is that if the environmental background shifts the whole distribution can "move." So societies or populations can get heavier in the aggregate but retain the same heritability. The median weight might vary but the rank order remains the same. The same can be said of height, but it likely there are more "natural" constraints in terms of how tall the average population may get before environmental inputs exhibit diminishing returns (in this way height may be more of a threshold trait where a "good enough" environmental background can result in expression of full genetic potential). Finally, I suspect that the exact nature of how genes effect height and weight differ as well insofar as I suspect there are more mediating factors in the latter than the former. David Kessler has recently been arguing that modern processed foods create a neurochemical response in the brains of humans which results in "food addiction." Whatever the truth of this particular claim, Kessler notes that some people seem to be able to control their impulses, or at least do not exhibit a strong fixation on particular tastes. He has suggested that there might be a heritable component here, and I think it gets to the likelihood that a lot of the heritability doesn't have to do with obesity or weight related genes like FTO, but with variation on cognitive traits which result in a propensity toward certain behaviors which result in weight gain. So despite the high heritabilities of both height and weight, I do think that the two traits are qualitatively different enough that a tight analogy may mislead. There are still strong class and region based differences in weight, while those for height have mostly disappeared in the United States. Addendum: I'm not addressing whether or not there is an "obesity epidemic." I'm not taking any sides as to why people are getting heavier, on average. Also, to be clear, I think a lot more of the range of weight is subject to gene-environment correlations than the range of height. To a far greater extent than height I believe weight has a strong biobehavioral component in terms of its determination.