Can a non-RE language be reduced to an RE language?
As you say, no non-RE language&nbsp;$U$ can be reduced to any RE language&nbsp;$L$, because, then, we could recognize&nbsp;$U$ using the reduction and the recognition algorithm for&nbsp;$L$.

The language&nbsp;$L_{diag}$ that you mention is indeed non-RE: it's the complement of an RE set that isn't recursive.  There is no recursive reduction from $L_{diag}$ to&nbsp;$L$.