The fact that "trade in the United States seems to have grown substantially faster than production" could be explained by reduction in trade costs. A decrease in importing partners' tariffs and non-tariff barriers (such as administrative fees or quotas) 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 and non-tariffs barriers.
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.