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I think that the answer to 3) is "No". Fo example, take two genus > 0 smooth curves over an algebraically closed field and glue them at a closed point. Then their pro-étale fundamental group $\pi_1^{BS}$ is the "Noohi completion" of the topological free product of the usual (profinite) étale fundamental groups of each of the curves $(\pi_1^{et}(C_1) *_{\mathrm{top}} \pi_1^{et}(C_2))^{\mathrm{Noohi}}$. I think that the sequences of specializations and generalizations should correspond to the elements of this (topological) free product, before taking the completions, and so should in general just have a dense image, and not give the entire group (the "free Noohi product" of two profinite groups is usually not profinite).