In equilibrium, if bonds are constantly being created and broken, where does the energy that does so come from?
The energy comes from the molecules themselves. See, they have certain energy of thermal motion. You might have heard of the ${3\over2}kT$ thing, but that's only an average value. Molecules have different energies, described by the so-called Maxwell distribution. Those with higher energy might take part in some reactions which are not possible for the entire bulk.

As for your second question, it's option 1: the equilibrium typically will shift. (Though the reaction might slow down a bit, too.)