To be honest the grounding of the 110 is good to do, however the trailer is not really grounded because of the tires...(reason to always use jack stands when parked)
Code says that the trailer neutral and ground should not be connected or bonded together for the AC system in the rig. That connection is only done at the service entrance where there is a proper earth ground. This means at the point the park attaches to the grid or at a genset (but for portable genset systems this is optional)
The tires and earth grounding the trailer has nothing to do with the safety precautions of the AC wiring system. It may be of some interest in the case of lightning but even there it doesn't make much difference.
You want to avoid multiple earth grounds in an electrical system because they can create what are called ground loops and other interesting phenomena. This is why you don't lie down on the ground but rather just crouch if you get caught out in a lighting storm.
For 110 the main safety feature of the ground is in the GFI or ground fault interrupter which detects any leakage between the power leads and ground and throws a breaker if that happens.
The ground in the plugs does allow all plugged in devices to have a common 'ground' reference and this should not be at any voltage difference from earth ground. Keeping any current from using the ground path is how that can happen.
Do not depend upon RV plugs, even at reputable parks, to be properly wired. That is why many of the older airstreams have this odd little light out on the rear skin so as to warn of a common miswiring. Melting the plug
has more on an example of an Oregon state parks wiring misery and things you should check each time you plug in.
For the 12VDC systems, the chassis is often used as a power conductor. This is to save on wire cost. As noted, 12VDC generally is not hazardous for human electrocution but you do need to exercise proper care and make sure to avoid shorts or loose connections that can create heat and maybe fire.
The chassis ground return path is a common source of 12VDC problems, especially between tow vehicle and trailer.
There should be no box in your rig that has both 12VDC wires and the 110VAC wires in it. If you look under the skin in your rig, you'll find that Airstream even separates the wiring paths as much as possible. AC over here and DC over there.
Also be careful about wiring color codes. There are some customs often used but the AC folks and the DC folks have different customs. When these meet, there are sometimes clashes. And then the repair folks get into the act with fixes that further confuse the issue. Always verify the purpose of the wire before assuming you know what it is connected to.