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Yes, this is true and follows from Wagner's theorem. Wagner's theorem asserts that every graph with no $K_5$ minor can be built from $0$-, $1$-, $2$, and $3$-sums from planar graphs and a fixed $8$ vertex non-planar graph called the Wagner graph. Since the Wagner graph is not $4$-connected, this implies that every $4$-connected graph with no $K_5$ minor is planar.

Alternatively, there is a short proof provided you are happy to assume Kuratowski's theorem. The proof idea is to start with a model of $K_{3,3}$ and then use $4$-connectivity to 'augment' the model of $K_{3,3}$ to a model of $K_5$. See the paper A Quick Proof of Wagner's Equivalence Theorem by Young.

Yes, this is true and follows from Wagner's theorem. Wagner's theorem asserts that every graph with no $K_5$ minor can be built from $0$-, $1$-, $2$-, and $3$-sums from planar graphs and a fixed $8$ vertex non-planar graph called the Wagner graph. Since the Wagner graph is not $4$-connected, this implies that every $4$-connected graph with no $K_5$ minor is planar.

Alternatively, there is a short proof provided you are happy to assume Kuratowski's theorem. The proof idea is to start with a model of $K_{3,3}$ and then use $4$-connectivity to 'augment' the model of $K_{3,3}$ to a model of $K_5$. See the paper A Quick Proof of Wagner's Equivalence Theorem by Young.

Yes, this is true and follows from Wagner's theorem. Wagner's theorem asserts that every graph with no $K_5$ minor can be built from $1$-, $2$, and $3$-sums from planar graphs and a fixed $8$ vertex non-planar graph called the Wagner graph. Since the Wagner graph is not $4$-connected, this implies that every $4$-connected graph with no $K_5$ minor is planar.

Alternatively, there is a short proof provided you are happy to assume Kuratowski's theorem. The proof idea is to start with a model of $K_{3,3}$ and then use $4$-connectivity to 'augment' the model of $K_{3,3}$ to a model of $K_5$. See the paper A Quick Proof of Wagner's Equivalence Theorem by Young.

Yes, this is true and follows from Wagner's theorem. Wagner's theorem asserts that every graph with no $K_5$ minor can be built from $0$-, $1$-, $2$, and $3$-sums from planar graphs and a fixed $8$ vertex non-planar graph called the Wagner graph. Since the Wagner graph is not $4$-connected, this implies that every $4$-connected graph with no $K_5$ minor is planar.

Alternatively, there is a short proof provided you are happy to assume Kuratowski's theorem. The proof idea is to start with a model of $K_{3,3}$ and then use $4$-connectivity to 'augment' the model of $K_{3,3}$ to a model of $K_5$. See the paper A Quick Proof of Wagner's Equivalence Theorem by Young.

Yes, this is true and follows from Wagner's theorem. Wagner's theorem asserts that every graph with no $K_5$ minor can be built from $1$-, $2$, and $3$-sums from planar graphs and a fixed $8$ vertex non-planar graph called the Wagner graph. Since the Wagner graph is not $4$-connected, this implies that every $4$-connected graph with no $K_5$ minor is planar.

Yes, this is true and follows from Wagner's theorem. Wagner's theorem asserts that every graph with no $K_5$ minor can be built from $1$-, $2$, and $3$-sums from planar graphs and a fixed $8$ vertex non-planar graph called the Wagner graph. Since the Wagner graph is not $4$-connected, this implies that every $4$-connected graph with no $K_5$ minor is planar.

Alternatively, there is a short proof provided you are happy to assume Kuratowski's theorem. The proof idea is to start with a model of $K_{3,3}$ and then use $4$-connectivity to 'augment' the model of $K_{3,3}$ to a model of $K_5$. See the paper A Quick Proof of Wagner's Equivalence Theorem by Young.