Are Context Sensitive Languages Turing Complete?
Context-sensitivity has nothing to do with the semantics of a language. A language could have an arbitrarily complicated grammar, and still not be able to do anything useful, perhaps because it has no looping construct.

To make the above possibly clearer, here's a simple example. We know that it is possible to list all Turing machines in some order. (The proof is not difficult and comes from Alan Turing himself). That means that we can associate every positive integer with some Turing machine, in such a way that every machine has a unique index. 

Now, take the language $L_1 = \{a^i \mid i &gt; 0\}$ (where $a^i$ means a string consisting precisely of $i$ $a$s). Since any string of $a$s is in $L_1$, the language is equivalent to the regular expression $a^+$.

The semantics of a program in  $L_1$ is simple: it does whatever Turing machine number $i$ would do, where $i$ is the length of the program. That's clearly a Turing complete regular language.

Then we define $L_2 = \{a^ib^i\mid i&gt;0\}$ and $L_3 = \{a^ib^ic^i\mid i&gt;0\}$, with the same semantics: in each case, program $i$ executes Turing machine number $i$. So these languages are also Turing complete, but $L_2$ is context-free and $L_3$ is context-sensitive.

Now, let's redefine the semantics. We'll define three languages with exactly the same syntaxes, but instead of executing Turing machine $i$, they print out $i$ in binary and then halt. These three new languages are still regular, context-free and context-sensitive, respectively, since the syntaxes have not changes. But none of them is Turing complete.

Bottom line: What a language means and how it is expressed are completely different concepts.