'87 Accord - automatic transmission replaced 30,000 miles/2 years ago.

When I went to start my car today, it happened...I got that high pitched
buzzing noise. It sounds a lot like the noise when you mitakenly start a car
when it is already running.

I've dealt with this before in another car, and I remember a mechanic
mentioning to me that a problem in the ring gear was to blame. I never had
it fixed - dismantling the transmission and replacing parts was cost
prohibitve, to say the very least. Anyway, I drove the car for quite a
while. The noise would appear intermittently, but after a couple of tries I
could always start the car. I observed no other problems in operating the
vehicle (nothing remotely related, anyway) once I got the engine running.

In my current vehicle, what other components could I or should I rule out
before I can confidently assume that the ring gear is a problem? Is the ring
gear part of the flywheel assembly, or can just the ring gear itself be
replaced?

Assuming ring gear replacement is prescribed, can a person just "let this
go" until the startup difficulties become more painful than the cost of
pursuing a solution, or is there likely additional wear and tear as a result
of the events and actions which make up the buzzing noise? Also, in addition
to whatever damage occurs during those few seconds of buzzing, does the
underlying problem go on to further damage/effect other components even
after the car has been started? My hope, of course, is that the needed
mechanical repairs will be exactly the same (parts and labor) whether I had
the issue fixed today or in a year or two or however long it would take
before reliably and quickly starting the car became an urealistic
expectation. Actually, I only need a couple of years out of this car and
then I can move on. I think the car's got those two years in it (excluding
the unknown ring gear factor). I'd like to just live with the occasional
horrible noise and a few additional seconds in turning over the motor, but
not if it is likely to make a serious dent in my two-year expectation.