Thank you for your thoughtful and generous response.
Questions:
So if I understand you correctly, I should ground the panel
chassis and then solder the XLR interconnect shells to the chassis?
Then, I should solder the shells of any XLR cables going into/out of
the panels to pin 1. Right?

Long Answers:
Yes to grounding/earthing the panel/metal enclosure, but No, and No only to
the outgoings from panel into studio.
Just ground the metal panel securely using a shakeproof washer/shakeproof
solder tag with decent, at least 15Amp, solid/stranded wire run to a
permanent, reliable earth/ground point. Nothing else needs to be done this
end.
The earth/ground point ought to be part of the building's tested and
approved wiring and should not be interruptible by plugging to obtain it!!
The add-on should be labelled "Safety Ground! DO NOT Remove."
[I may sound like I am teaching my grandmother to suck eggs - but I don't
know the relevant bye-laws in your country.]

Explanations
Any shells of metal XLRs plugged into the panel metal-flanged receptacles
will now make to chassis-ground by default. Simple! If the panel connectors
have
insulated barrels and wells, on the types I've used, they still had a "shoe"
that makes with an inserted shell.
All Pins 1 will go their own route as screens to/from technical equipment
which is
hopefully and advisedly grounded during use and again hopefully and
advisedly during inactivity
(just switch off at wall outlets but without unplugging relevant mains
cords).
NOTE Pin 1 at XLR *female* end (outgoing from mic) of a studio wander cable
would be linked to
shell tag as it is often handheld or manhandled.
BUT regarding incoming feeds from control room equipment on roving male XLRs
at the studio floor end,
it remains optional for the user to link /isolate the shell to /from its pin
1, dependent on *no-hum results* with mains-operated folback/heaphone/comms
amps, etc.
Remember, all exposed metalwork on mains appliances should be
grounded/earthed by their cords (well, that's the regulation over here, at
least).

Q
Can I connect the chassis plates to the ground on a nearby AC line or
do I have to run a separate line?

Answer:
As I've said, run it separate, not from plug-in cords - so making it
permanent at each end.
Of course, these are only safety routes to neutralise potential danger to
manhandled hardware, no
current is involved [till a possible live mains calamity is prevented by
shorting to ground].

Q
Can I connect both junction faces to the same ground?

Answer:
Yes, several even, if the captive screw is long enough to accommodate the
new wires securely between disc washers and a locknut (or via brass ground
bus connector terminals). This paralleling is normally called a star
connection.

Hint. Choose the shortest grounding/earthing route possible to good and
proper electrical infrastructure. Ideally anchor it at a mains consumer
unit, or it's acceptable at
a ring-main outlet node, but avoid grounding at a spur outlet box where
resistance is slightly higher.