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You can't linearly order the Vitali ($\mathcal{P}(\omega)/\mathrm{Fin}$) degrees if every set of reals has the property of Baire, since you can't even choose between complementary degrees. There's a map that sends mod-finite different subsets of $\omega$ to mutually Cohen-generic reals over $L$ (assuming $\mathcal{P}(\omega)^{L}$ is countable; this is overkill in any case), giving you an embedding of the Vitali degrees into the Turing degrees. This should answer the first question.