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Wikipedia states the definition of Fermi energy as for "a system of non-interacting fermions". If we have to assume free electrons in a solid behave this way before we are able to calculate Fermi energy, how can Pauli exclusion be justified (because electrons are non-interacting)? Can Fermi energy be similarly defined for electrons confined to a single atom?

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You might say, of course, that in some sense the fermions do interact (and it is even called exchange interaction). However, it is physical forces, like Coulomb ones, that are understood to be absent. A relevant discussion has taken place here: Degeneracy Pressure, What is it?

Concerning the second question about single atoms, the answer is no. Firstly, Fermi statistics, as any statistics, can only be applied to macroscopic objects. Secondly, even if you were able to create a giant nucleus of a large charge and cover it with a macrosopic number of electrons, the electrons would be interacting with each other through Coulomb forces, hence will not represent a degenerate gas and hence will neither follow the statistical distribution of degenerate gases nor possess Fermi energy.

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