Why will world population keep growing if all women have only 2.1 children
As you mention, the population-lag effect is responsible for this. From The Wikipedia article on TFR: http://en.wikipedia.org/wiki/Total_fertility_rate#Population-lag_effect


  A population that has recently dropped below replacement-level
  fertility will continue to grow, because the recent high fertility
  produced large numbers of young couples who would now be in their
  childbearing years.


Thus, even if the current TFR implies long-term stability, the recent history of TFR values will continue to affect birth rates, and thus population growth/decrease, in the future.

Imagine that you have a stable population, with static birth and death rates. Now imagine that the birth rate during a single year for some reason doubles before dropping back to normal the next year, thus transiently increasing TFR. Now, it is easy to see that when the newly born, larger-than-normal generation reaches reproductive age, the population will increase again because the birth rate will increase in proportion with the size of the generation, while the death rate is unaffected until the enlarged generation reaches old age and starts dying off. Later, the population will decrease to its long-term stable level, which will be larger than before the TFR spike.