Are empty-set languages recursively enumerable?
The empty language is recursive. The TM accepting this language just returns $0$/REJECT on any input. Similarly, the language $\Sigma^*$ is also recursive. Just return $1$/ACCEPT on any input.  

Basically, it does not matter how a language is defined. What makes them recursive/RE is the fact that we can effectively decide its membership. For 
a recursive language $L$ there is a TM that always accepts $x$ if $x \in L$ and rejects $x$ if $x \notin L$. For a recursively enumerable language $L$  there is a TM that always accepts $x$ if $x \in L$.