Is there a stutue of limitation on how long a doctor or other
healthcare provider can wait to submit outstanding bills?

depends on which insurance carrier you had/have, if provider needed to
submit tx verifications or you needed authorization/precert, lots of
reasons. Many carriers request HICFA/other submission within 3 months.
If provider is late submitting, the carrier can say no and you have to
go thru appeals process. You can get stuck if 'fine print' on consent
forms you signed indicate you are responsible.

This depends a lot on the jurisdiction, and probably the individual
carrier. For example, in Minnesota, carriers may reject claims that
have not been submitted within 15 months. Carriers might accept older
claims if
A) Medicare is involved, or
B) There is a really good reason, for example if the patient is
deceased or incapacitated.

Liability for refused charges might rest with either the subscriber or
the provider, depending on the type of relationship the provider has
with the carrier, or as the other comments notes, what the subscriber
signed.