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I guess you are confused by the action on wave functions and operators. 1) The second approach to your first concern is "more correct." Notice that $t(k)$ or $w(k)$ is a $c$-number( or scalar, not an operator). When you write $w(k) \rightarrow w(-k)$, I guess you treat it as a wave function (I am not sure, because if it is a wave function, it should ...
The Hamiltonian is time-reversal invariant: $c_{k\uparrow}\rightarrow c_{-k,\downarrow}, c_{k\downarrow}\rightarrow -c_{-k,\uparrow}$. You can check that explicitly. The ground state is also invariant, because Cooper pairs are all spin singlet. One of the significant implications of time-reversal symmetry for s-wave superconductors is the Anderson's ...