Suppose that $T$ is a consistent first order theory. Now let the language of $T$ be $L\_T$. 


**Question:** is it always consistent to add a new primitive constant $D$, and a new primitive binary relation $\in^\*$ called 'class membership', and add a new symbol $\epsilon$ and axiomatize that for each formula $\phi$ the string $\epsilon \phi$ is a term of the language, and add the following schema to all axioms of $T$ bounded $\in^\* D$


If $\phi$ is a formula in $L\_T$ in which all and only $y,w\_1,..,w\_n$ occur free, and only occur free, and if $\phi^D$ is the bounded $ \in^\* D$ form of the formula $\phi$, then:


$ \forall w1,..,wn \in^\* D \ \forall y (y \in^\* \epsilon (\forall y\phi^D )\leftrightarrow \phi^D)$,


is an axiom.



Can we have a nearily unrestricted class comprehension over predicates that do not mention the class membership symbol
If I understand you correctly, the answer is yes, providing that you don't just add those axioms directly to $T$, but instead add the assertion $\phi^D$ for each axiom of $T$. 


Your theory is simply the syntactic analogue of taking a model $M$ in the language $L\_T$, and adding a second-order part with an object representing each definable subset of $M$ (allowing parameters). Thus, the extension of $D$ is simply the original model $M$; the interpretation of $\epsilon\phi$ is the function mapping the parameters to the class defined by this formula, and so on. This natural interpretation of your theory shows that any model $M$ of $T$ can form the $D$-part of a model of your new theory. This argument also shows that the new theory is conservative over $T$ for assertions in the language of $L\_T$ relativized to $D$. 


