How exactly does etching using NaOH work when synthesizing materials with SiO2 templates, and does etching time or etching concentration have any effect?
SiO2 can be viewed as the anhydride of the silicic acid Si(OH)4. However, in acid medium, neither the acid or any associated ion is stable, so it would drop the water and crystallize. That's why you can store acids in glass bottles safely. The only exception here is HF because it replaces an Si-O by an Si-F binding situation which is comparably stable.
In alcalic medium, on the other hand, silicates have a certain degree of solubility and that's why OH- is capable of etching glass. This pH dependency can be seen in the upper graph here:
Regarding the last part of your question: since the etching process is not super effective at room temperature yet, there is of course a large dependence on time and concentration. Please be aware that the crystallinity of the SiO2 also plays a huge role, so amorphous oxide will be etched a lot faster than a sintered crystal, so be careful when you compare your own results to literature data.