How does one solve this mechanism? Why doesn't the negatively charged oxygen on the non-cyclic molecule remove an alpha hydrogen from the cyclic one?
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It certainly could undergo the acid-base equilibrium you described, if there's enough energy in the system. Your problem is omitting an important piece of information, which is the temperature at which the reaction is taking place. Lithium enolates (the right compound) are usually formed at low temperatures, down to -78 C. The hope/assumption is that the aldol reaction happens faster than the acid-base reaction between the two reactants. This is certainly hard to control in this particular example, and thus the yield of the reaction is not great, according to this example mentioned in a review article on aldols:
Furthermore, it's likely that in doing this reaction, the experimentalist would form the lithium enolate separately before mixing it with the cyclohexanone. You can read about the importance of order of addition here: https://en.wikipedia.org/wiki/Aldol_reaction#Order_of_addition
Here's the mechanism, note that the problem didn't provide you with a viable source of H+.