I notice that US exports doubled between 2000 and 2011, imports also increasing greatly, but GDP only grew by 20%. Thus, trade in the United States seems to have grown substantially faster than production. Why is this?
The fact that "trade in the United States seems to have grown substantially faster than production" could be explained by reduction in trade costs. For instance, a decrease in importing partners' tariffs tends to favor US exports. So, for the same level of production or GDP, US sales will be more export oriented. It works similarly for US imports following a reduction of US tariffs.
A first explanation to the reduction in trade costs is the reduction of tariffs and non-tariff barriers (such as administrative fees or quotas), that is changes in trade policy. Trade liberalization policies could be unilateral, multilateral, regional or bilateral. Regional trade agreements for instance have become increasingly prevalent since the early 1990s, according to the WTO.
A second explanation to the reduction in trade costs is related to technological progress such as faster communications.
Note that it is common to compute a country's trade openness index as the ratio of a country's trade (exports plus imports) to its GDP. Controlling for GDP allows economists to evaluate the outward or inward orientation of a given country's economy. Moreover, this ratio could be computed over time to measure the extent of globalization. For instance, according to the World Bank data Chinese trade openness ratio raised from 2.6% in 1970 to 39% in 2006. The US ratio increased from 9.7% in 2001 to 13.5% in 2011, which is consistent with the facts you mentioned.