reconnect the faraday shield to mains ground (the lug on the chassis).
--- Yes this may reduce some noise from and to the mains line.
Leave the "neutral to ground" connections disconnected as shown.
--- Remove the wire entirely or route it to a mounted sheathed banana jack to enable easy jumper to ground if you ever find a need, don't leave bare ended wires flopping around. I haven't seen a true medical grade isolation transformer that has one of the output leads grounded; that would defeat most of the purpose of using an isolation transformer. On my similar transformer I connected each output line to a red sheathed 4mm banana jack and covered the openings with vinyl caps to keep pokey bits out. This allows for easy connection to a meter or something else.
I did check the outlet ground with an ohm meter and they are grounded to the chassis.
--- This is very standard for North American wiring to have the U ground terminal of the socket connected the framework of the socket which in turn may be connected to a metal enclosure via the mounting screws and the center screw between the two parts of the duplex outlet will also connect to the framework providing a ground if a metal cover plate is used.
If I change the outlets to "isolated ground (orange) outlets", do I leave that ground lug unterminated?
--- If you have to ask, you probably shouldn't. You have to think long and hard as to why, if ever, you would want the chassis of your DUT not grounded. If the actual need ever arises, there is always the option of using one of the two to three prong adapters -- but be aware that it is easy to forget. I would absolutely not install an isolated ground receptacle and leave it ungrounded. Note that on my own isolation transformer I connected the isolated ground for the receptacle to the green / yellow binding post adjacent to the green binding post with an easy to connect jumper bus. It is very easy to see and hard to forget. The green binding post is connected to mains ground and the transformer case.
(Isolated ground receptacles are not all orange, they are available in many colors. Look for the green triangle on the face that indicates isolated ground. Medical grade isolated ground receptacles often are white or clear.)
Isn't that the same result that I get in my pic by disconnecting the ground wires from the chassis?
--- No, the metal transformer enclosure is still connected to the outlet via the mounting screws; by all means don't isolate the transformer enclosure from mains ground.
If they get wired back to mains ground then doesn't that defeat the purpose of the isolated ground outlet?
--- A better question is: Why would you want or need your ground to be isolated? Probably not for whatever you will be doing with your isolation transformer. Medical installations install a totally separate ground grid to hopefully mitigate any possible transient currents that may be on the mains ground wiring due to a fault to ground somewhere in the system. For instance if you are working with RF and have installed a system of ground rods around your house to connect your isolated ground, it may help to use that ground ring isolated from the mains ground.
Remember Murphy's Law is waiting in the wings to hand out Darwin awards.