The problem is you are talking about orthogonal concepts, ie. economic policy and social policy. See the Political Compass: https://www.politicalcompass.org/

Not really, no, since all these mentioned political philosophies have explicit market elements.

Or do you think it’s possible to have a socialist or communist gov’t AND capitalism or free market at the same time?

Because those quite literally are incompatible as part of the definition.

Same as a heavily regulated economy or state ownership of production in Anarchism.