Truly Understanding the Second Law of Thermodynamics and Entropy
Be careful in citing equilibrium as an example, as it fully integrates entropy in its definition. See this article for the details, but in a spontaneous reaction the entropy is increasing, and part of reaching equilibrium is that the energy gain and loss among products and reactants is balanced. Also, most experimental reactions are (nearly) closed systems for short periods of time, and if no energy is gained or lost (again, for short periods), you can have an equilibrium where there is no change in entropy. If you let your experiment sit and the heat of the system changes (for example), the equilibrium will change as well. And eventually you end your experiment, clean out your flask, and thus increase the entropy of the universe again!

@MARamezani's comment pointing to the article on Biology looks like a good answer to your biology questions. Earth is most definitely not a close system with that giant energy source 93 million miles away.