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Huh? What's wrong with measuring in price/watt? Price/watt is actually the standard for comparing electricity generation due to not having to create arbitrary units of time to compare with

You're mistaken. Fuel is sold by price/energy, effectively, but generating capacity is sold by price/power. Since in the case of solar PV, the "fuel" is free, the only cost worth mentioning is the capacity cost.

Both are relevant. The government and companies probably build capacity in watts (power), but consumers purchase actual joules (energy).

Installed capacity is normally measured in watts. It's really only valuable for trending within the same technology, though, unless you include the capacity factor.

No, it shouldn't be Ws. It's the ratio of the manufacturing cost per rated power of PV-modules.