I believe the answers to these questions are all positive. This kind of problem was discussed by Groszek and Laver in Finite groups of OD-conjugates [Period. Math. Hungar. 18 (1987), 87-97, MR0895774]. Answering a question of Mycielski, they show that there can be two sets of reals $x,y$ such that $\lbrace x,y\rbrace$ is ordinal definable but neither $x$ nor $y$ is ordinal definable. They also prove a lot of other interesting things about OD conjugates.
Here is the brief argument from the intro to that paper. Suppose $u, v$ are two mutually Sacks generic reals over $L$. Both $u$ and $v$ have minimal degree over $L$ and they satisfy the same formulas with ordinal parameters since Sacks forcing is homogeneous. L$. Let$x$and$y$be the$L$-degrees of$u$and$v$respectively. Then$x$and$y$also satisfy the same formulas with ordinal parameters because Sacks forcing is homogeneous. However,$\lbrace x, y \rbrace$is definable (without parameters) since these are the only two minimal$L$-degrees in$L[u,v]$. 2 added proof; added 29 characters in body I believe the answers to these questions are all positive. This kind of problem was discussed by Groszek and Laver in Finite groups of OD-conjugates [Period. Math. Hungar. 18 (1987), 87-97, MR0895774]. Answering a question of Mycielski, they show that there can be two sets of reals$x,y$such that$\lbrace x,y\rbrace$is ordinal definable but neither$x$nor$y$is ordinal definable. They also prove a lot of other interesting things about OD conjugates. Here is the brief argument from the intro to that paper. Suppose$u, v$are two mutually Sacks generic reals over$L$. Both$u$and$v$have minimal degree over$L$and they satisfy the same formulas with ordinal parameters since Sacks forcing is homogeneous. Let$x$and$y$be the$L$-degrees of$u$and$v$respectively. Then$x$and$y$also satisfy the same formulas with ordinal parameters. However,$\lbrace x, y \rbrace$is definable (without parameters) since these are the only two minimal$L$-degrees in$L[u,v]$. 1 I believe the answers to these questions are all positive. This kind of problem was discussed by Groszek and Laver in Finite groups of OD-conjugates [Period. Math. Hungar. 18 (1987), 87-97, MR0895774]. Answering a question of Mycielski, they show that there can be two sets of reals$x,y$such that$\lbrace x,y\rbrace$is ordinal definable but neither$x$nor$y\$ is ordinal definable. They also prove a lot of other things about OD conjugates.