Fixed points in dependent type theories
I think the idea of unifying type and term-level fixpoints is natural, though I have to admit I'm not sure that reducing the number of constructions of a system is not always a recipe for conceptual simplicity.

One source I have for a system that takes this approach is the $\Pi\Sigma$ system:

$\Pi\Sigma$: Dependent Types Without the Sugar, Altenkirch, Danielsson, Löh and Oury.

You might also want to look at these slides by Khulmann on equi-recursive types versus iso-recursive types: equi-recursive types embody the kind of type-level recursion you are interested in (iso-recursive types have explicit folding and unfolding). As you suspect, type-checking with equi-recursive types is tougher (but not impossible!).

As a final remark, adding equi-recursive types to an otherwise consistent system does not make it inconsistent, if positivity constraints are respected.