Frank wrote:
2) Should a "in-glass" rear winshield antenna be grounded to the
car's body? If so, from which end of the coax cable? (see Q 3)

Grounding an antenna completely defeats its purpose.
Is this also part of a defrosting heating element or such?

3) Does the expression "grounding the antena" (in the case of a
glass mounted antenna) be as simple as soldering a grounding wire
from the outside coax lead end that plugs into the HU to a metal
ground of the car? Or should it be done closer to the actual
antena? SUch as at the junction of the end of the coax and where
the antenna starts...

The expression "grounding the antenna" is nonsense
in the absense of better colatteral information. If it is
just an antenna (and not part of a heater or something)
then "grounding the antenna" sounds stupid.

What/who/why is "HU"? If this is a part of the circuit
you are working on, you need to reveal the entire situation.
Otherwise, you will get misleading responses based on
guessing from thousands of miles away.

Here is more detail on my problem if this helps: inconsistent
FM reception ( static) with in-glass antenna where I use
to get crystal clear reception with old car.

In-glass antennas have never been noted for their good
performance. They are only popular because they are
easier for the auto manufacturers (and car washes). Many
people add a proper antenna after-market when they are
stuck with these things.

However, there is a twist to my problem: I noticed that
the coax cable (inner cable) from the HU is not truly
connected to the antenna even if it looks like it is. I tested
it with a continuity tester and it is not.

If "HU" means Heating Unit or something, it seems likely
that the antenna lead is connected via a capacitor. The cap
would conduct RF voltage while appearing to be an open
circuit at DC (which is what your meter is using.)