This is a common issue in Austronesian linguistics where the notion of precategorial (=functionally unspecified) roots is often employed to explain the fact that roots don't have a POS category until they're employed in an utterance, and then the same root can be used in many different POS categories. This may match the situation you describe, where the root ...
Yes, some linguists consider this possible. Here are some such concepts/authors:
Pesetsky, David. 1995.Zero syntax: Experiencers and cascades (CurrentStudies in Linguistics 27). Cambridge: The MIT Press.
Borer, Hagit. 2005. Structuring Sense Volume I. Online:http://dx.doi.org/10.1093/acprof:oso/9780199263905.001.0001.