Truly Understanding the Second Law of Thermodynamics and Entropy There are some seemingly contradictory things I’d like cleared up:

Second Law states: $\mathrm dS_\text{univ} = \mathrm dS_\text{sys} + \mathrm dS_\text{surr}$, and $\mathrm dS_\text{univ}$ is always positive in a spontaneous process.

First off, what does “spontaneous” truly mean? “It means the reaction occurs without any external energy input” – I’ve heard this one before, but it doesn’t make sense. In an equilibrium there is a spontaneous direction (the side that is favored) but the reverse reaction also occurs, though at a slower rate, than the spontaneous reaction. What is the most fundamental definition, intuitive and universally applicable definition of spontaneous?

Moving on, according to ChemWiki the Second Law states that the entropy change of the universe can never be negative, even though my textbook says that this is only true for spontaneous processes. Who is right and why (explain in quantum terms, since I understand that in the long run of course the entropy of the universe increases by the laws of probability)?

An example to work through:

In a living cell, large molecules are assembled from simple ones. Is this process consistent with the second law of thermodynamics?

Book’s solution: To reconcile the operation of an order-producing cell with the second law of thermodynamics, we must remember that $\mathrm dS_\text{univ}$, not $\mathrm dS_\text{sys}$, must be positive for a process to be spontaneous. A process for which $\mathrm dS_\text{sys}$ is negative can be spontaneous if the associated $\mathrm dS_\text{surr}$ is both larger and positive. The operation of a cell is such a process.

First off, are we to consider the operation of a cell in this problem spontaneous? As I recall from biology, life processes require ATP, the carrier of energy, so these would technically “require external energy input,” would they not? However, the book treats it as a spontaneous process. Again, this confuses me because the definition of spontaneity is inadequate as I now semi-understand it.

Sorry for such a long post, but these are very important and fundamental ideas in spontaneity, entropy, and free energy, and I don’t want to read on until these ideas are well ingrained, and correctly so.
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.