One way of avoiding such paradoxes is weakening the base logic that we use it for reasoning in our formal theory. One of these logics is Visser-Ruitenburg Basic logic. This logic is a sub-intuitionistic logic which satisfies this equation $\bf \frac{IPC}{S_4}=\frac{BPC}{K_4}$ with respect to translation of Godel (intuitionistic logic to modal logic). By removing modus ponens from $\bf IPC$ and adding some rules and axioms like cut, transitivity and etc to logic, $\bf BPC$ is defined. So it is not true that ${\bf BPC}\vdash \top \to \phi \Rightarrow \phi$ for every formula $\phi$.

Maybe this answers to your first question that Fregean Set theory or ${\bf FST}$ is consistent and has Kripke models with respect to this logic. I'm not sure, but I think that $\bf BPC$ is maximal logic that $\bf FST$ is consistent with respect to it.

References

1. Basic propositional calculus I, Mathematical Logic Quarterly 44 (1998), pp. 317--343.
2. Basic predicate calculus, Notre Dame Journal of Formal Logic 39, No. 1 (1998), pp. 18--46.
3. Basic logic and Fregean set theory, in H. Barendregt, M. Bezem, J.W. Klop (editors), Dirk van Dalen Festschrift, Quaestiones Infinitae, Vol. 5, Department of Philosophy, Utrecht University, March 1993, pp. 121--142.
4. Constructive logic and the paradoxes, Modern Logic 1, No. 4 (1991), pp. 271--301.

One way of avoiding such paradoxes is weakening the base logic that we use it for reasoning in our formal theory. One of these logics is Visser-Ruitenburg Basic logic. This logic is a sub-intuitionistic logic which satisfies this equation $\bf \frac{IPC}{S_4}=\frac{BPC}{K_4}$ with respect to translation of Godel (intuitionistic logic to modal logic). By removing modus ponens from $\bf IPC$ and adding some rules and axioms like cut, transitivity and etc to logic, $\bf BPC$ is defined. So it is not true that ${\bf BPC}\vdash \top \to \phi \Rightarrow \phi$ for every formula $\phi$.

Maybe this answers to your first question that Fregean Set theory or ${\bf FST}$ is consistent and has Kripke models with respect to this logic. I'm not sure, but I think that $\bf BPC$ is maximal logic that $\bf FST$ is consistent with respect to it.

References

1. W.Ruitenburg, M.Ardeshir, Basic propositional calculus I, Mathematical Logic Quarterly 44 (1998), pp. 317--343.
2. W.Ruitenburg, Basic predicate calculus, Notre Dame Journal of Formal Logic 39, No. 1 (1998), pp. 18--46.
3. W.Ruitenburg, Basic logic and Fregean set theory, in H. Barendregt, M. Bezem, J.W. Klop (editors), Dirk van Dalen Festschrift, Quaestiones Infinitae, Vol. 5, Department of Philosophy, Utrecht University, March 1993, pp. 121--142.
4. W.Ruitenburg, Constructive logic and the paradoxes, Modern Logic 1, No. 4 (1991), pp. 271--301.