Was the Renaissance a cause of rapid population increase from 1400 AD?

Answer 1

No, except perhaps later and indirectly. It was a cultural and intellectual movement and as such didn't impact materially on fertility or mortality except among the few who might benefit from medical progress. Population growth didn't resume generally in Europe until the latter half of the 15th century, and at an average annual rate of less than 0.3% over the next 150 years it's questionable if it can be considered rapid. The reasons include a (still not understood) lessening of the mortality associated with epidemics, gradual agricultural improvement and ongoing social change which encouraged production and opportunity, and from the sixteenth century (though mostly from the eighteenth) the introduction of new crops from the Americas.