1. They are all residually finite.
2. They are not all subgroup separable/LERF.
3. They do not all have decidable submonoid membership problem.

Residual finiteness is a result which can be found in the (aptly named) [G. Baumslag, "Finitely generated cyclic extensions of free groups are residually finite" (Bull. Amer. Math. Soc., 5, 87-94, 1971)].

The facts on the submonoid membership problem and subgroup separability/LERF follow from the following example: the free-by-cyclic one-relator group $$G = \langle a, t \mid [a, tat^{-1}] = 1 \rangle \cong \langle a, b, t \mid a^t = ab, b^t = b \rangle$$ embeds the right-angled Artin group $A(P_4)$ (see this article), which is known to have undecidable submonoid membership problem. It is shown that $G$ is not subgroup separable/LERF in [R.G. Burns, A. Karrass, and D. Solitar, A note on groups with separable finitely generated subgroups, Bull. Aust. Math. Soc. 36 (1987), 153–160].