Can non-recursively enumerable languages be solved with dovetailing?
Dovetailing is just a technique for programming Turing machines. It doesn't give Turing machines any additional power. In particular, dovetailing cannot cause a Turing machine to decide an undecidable problem, recognize an unrecognizable language, and so on.

While dovetailing might seem like an arcane technique, it is actually very similar to context-switching, which enables our computers to run many programs in parallel. This doesn't give our computers any more power – on the contrary, context-switching incurs overheads.