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I am given the substrate and reagents in steps (1,2,3). I am asked to figure out the product! I am not sure how the reagents in 2 (aq. Na2CO3) and 3 (NaH, MeI/DMF) work to give that answer. Why is the answer what it is?

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You can work backwards here. Step 3 converts cyclohexanol to the product by substitution reaction of methyl iodide and alkoxide. Sodium hydride is the base that deprotonates the alcohol.

Step 2 is a bit dubious. They want to substitute a secondary chloride with hydroxide. In principle this will work, though I am suspicious about whether or not it will actually take place. For cyclohexyl chloride (product from step 1), the substitution is probably only going to happen from the axial conformer; the equatorial conformer has the flag pole hydrogens blocking the approach to the $\sigma^{*}$ orbital. But from the axial conformer, I suspect that elimination is also a competitive pathway. Hard for me to say exactly what the relative rates are for sure.

Step 1 you've likely figured out. It's radical chlorination of an alkane.

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