This class is the same as the $\Delta$-confluent graphs introduced by Eppstein, Goodrich, and Meng (Delta-confluent Drawings). It is easy to see that $\Delta$-confluent graphs are closed under the operations you allow (and a single vertex is $\Delta$-confluent). On the other hand, a $C_5$ in a $\Delta$-confluent drawing must use a $\Delta$-junction, since the tree-confluent graphs are all bipartite (see Hui, Pelsmajer, Schaefer, Stefankovic, Train Tracks and Confluent Drawings). But then there must be either two branches of the $\Delta$-junction each containing two vertices of the $C_5$ or one branch that contains three vertices of the $C_5$. In either case, the $C_5$ must have two disjoint chords.

Eppstein, Goodrich, and Meng also prove that the $\Delta$-confluent graphs are the same as the distance-hereditary graphs, so the result follows from that.

This class is the same as the $\Delta$-confluent graphs introduced by Eppstein, Goodrich, and Meng (Delta-confluent Drawings). It is easy to see that $\Delta$-confluent graphs are closed under the operations you allow (and a single vertex is $\Delta$-confluent), and all $\Delta$-confluent graphs can be built using these operations (easy induction).

Eppstein, Goodrich, and Meng also prove that the $\Delta$-confluent graphs are the same as the distance-hereditary graphs, so the result follows from that.

Theorem 2 in the Eppstein, Goodrich, and Meng paper is the result tying distance-heredity to the operations you allow. It's attributed to a 1986 paper by H. Bandelt and H. M. Mulder. Distance-hereditrary graphs.

This class is the same as the $\Delta$-confluent graphs introduced by Eppstein, Goodrich, and Meng (https://arxiv.org/abs/cs/0510024). It is easy to see that $\Delta$-confluent graphs are closed under the operations you allow (and a single vertex is $\Delta$-confluent). On the other hand, a $C_5$ in a $\Delta$-confluent drawing must use a $\Delta$-junction, since the tree-confluent graphs are all bipartite (see Hui, Pelsmajer, Schaefer, Stefankovic, https://link.springer.com/article/10.1007/s00453-006-0165-x). But then there must be either two branches of the $\Delta$-junction each containing two vertices of the $C_5$ or one branch that contains three vertices of the $C_5$. In either case, the $C_5$ must have two disjoint chords.

Eppstein, Goodrich, and Meng also prove that the $\Delta$-confluent graphs are the same as the distance-hereditary graphs, so the result follows from that.

This class is the same as the $\Delta$-confluent graphs introduced by Eppstein, Goodrich, and Meng (Delta-confluent Drawings). It is easy to see that $\Delta$-confluent graphs are closed under the operations you allow (and a single vertex is $\Delta$-confluent). On the other hand, a $C_5$ in a $\Delta$-confluent drawing must use a $\Delta$-junction, since the tree-confluent graphs are all bipartite (see Hui, Pelsmajer, Schaefer, Stefankovic, Train Tracks and Confluent Drawings). But then there must be either two branches of the $\Delta$-junction each containing two vertices of the $C_5$ or one branch that contains three vertices of the $C_5$. In either case, the $C_5$ must have two disjoint chords.

Eppstein, Goodrich, and Meng also prove that the $\Delta$-confluent graphs are the same as the distance-hereditary graphs, so the result follows from that.