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. The set of $x$ which are chosen can't be meager or somewhere comeager, since below any initial segment you can find a pair of complements which are both in any given comeager set. 

There's a continuous, Vitali-invariant map $c$ 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. Such a map can be induced by a pair of functions $f, g \colon \omega \to \mathrm{Fin}$ so that for all $x \subseteq \omega$, $c(x) = \bigcup\{ f(n) : n \in x\} \cup \bigcup \{ g(n) : n \not\in x\}$. This should answer the first question.