I have interpreted the exercise in the following way: We are working in ZFC. In particular, this means that any element of a set is also a set. I have looked on the relation $\in$ for some simple sets which can be obtained from the empty set $\emptyset$.
When I look at the table of contents of Devlin's Joy of Sets (Fundamentals of Contemporary Set Theory), I saw that the author deals with axiomatic set theory only in the second chapter of this book. So it is possible that I have somehow misinterpreted what he wanted in this exercise.
Let us start by two trivial observations. If $x=\emptyset$, then all properties are vacuously true. If the set $x$ has one element, then $R$ is connected and antisymmetric, since we never have $a\ne b$.
The relation $\in$ on any set $x$ must be antisymmetric, since $a\in b\in a$ contradicts Axiom of regularity.
If $x=\{\emptyset,\{\emptyset\}\}$ then the relation $\in$ is not symmetric, just take $a=\emptyset$ and $b=\{\emptyset\}$. Notice that relation $\in$ on this set is connected and it is also transitive.
Let $x=\{\emptyset,\{\{\emptyset\}\}\}$. If we denote $a=\emptyset$ and $b=\{\{\emptyset\}\}$, then neither $a\in b$, nor $b\in a$. So the relation $\in$ on this set is not connected. However, as it is the empty relation, it is transitive.
Let $x=\{a,b,c\}$ where $a=\emptyset$, $b=\{\emptyset\}$ and $c=\{\{\emptyset\}\}$. Then $a\in b$ and $b\in c$, but $a\notin c$, so the relation $\in$ on this set is not transitive. It is also not connected, since neither $a\in c$ nor $c\in a$.
Best Answer
The second relation is not a partial order, because it fails antisymmetry, but it is an equivalence relation (because it is reflexive, symmetric, and transitive, but not anti-symmetric, for the reason you state: $aRb$ and $bRa$, but $a \neq b$).
The first relation is both an equivalence relation and, as you note, a partial order. (It is vacuously symmetric and antisymmetric and transitive).
Note: A set, e.g., $A= \{a, b, c\}$ is such that there is no duplicity of elements. So, e.g., you can rest assured that $a \neq b$, else the set would be $A= \{a, c\},$, or $A= \{b, c\}$. By the definition of a set, every listed in the set, is distinct wrt other elements in the set.