How to properly ground chassis containing AC/DC switching power supply

Is there any danger in bonding the DC- output to Ground with a short wire, to ensure my circuit ground is also referencing the home ground? I'm not sure simply letting the outputs float is wise,

Assuming the power supply is properly built either is fine.

Side-note: I have an after-market laptop supply that's two-prong... it works either way, but if I plug it in one way the metal trim on the laptop has an interesting "buzz" to the touch. Not shocking, but definitely noticeable. I suspect that when it's plugged in the right way the output ground is weakly referencing neutral, and the "buzzy" way has the output ground weakly referencing line voltage (it's floating - it'd be quite a bit more than buzzy otherwise). The OEM supply with the 3-prong plug? Fully grounded to the shield of the DC power cord.

Power supplies without an earth connection often have problems like this, especially as they get larger.

Capacitors have to be placed between input and output to control EMI. In an unearthed design this ends up with the output being "weakly referenced" to the input. How weak that reference is (or to put it another way how big the "touch current" is) depends on the size of the capacitors. Unfortunately there is a tricky compromise here, bigger caps are better at suppressing EMI but produce larger "touch currents". Larger power supplies tend to suffer worse from this than smaller ones (this is why laptop PSUs from reputable brands are usually earthed while phone chargers are usually not).

In an earthed design this can be mitigated either by tying the output to ground (pretty much universal in in desktop PC power supplies, occasionally seen in laptop supplies) or by splitting the EMI suppression capacitors into two parts, one from output to earth and one from input to earth.

Why is the DC ground connected to mains earth? While I am not overly familiar with safety qualifications, I believe it is now mandatory, and has been "best practice" for quite some time, to have some potential in the system tied to earth, and typically to the chassis. As long as the DC output is not required to be floating, the most common connection is earth/chassis/DC ground all tied together. This output is generally the one of use to the human being utilizing the supply. This is different than say a lab/bench supply where you typically have a "-" prong that is the floating supply's return path and a GND/Earth prong that is tied to chassis and earth for safety reasons. This allows you to stack supplies which you can't do in the case of a earth-referenced output.

As for the LED, you are exactly correct. But, if chassis = DC GND = earth, this works. Whether or not that is allowable by modern-day safety standards is unknown to me. This isn't that much different than how an automobile is typically wired, and I'm sure that wasn't completely uncommon "back in the day". I believe that CRT TVs/monitors still had live chassis connections even into the 90s.

I don't feel very comfortable answering the last part of your question. If this is an isolated supply with earth connection on the DC/output side, the AC side has to be considered completely floating. Don't go sticking a probe in there with your ground clip on the DC side. Tektronix did a very good whitepaper on how to probe floating supplies several years back. I highly recommend you try to track that down. I believe it was called Fundamentals of Floating Supply Measurement. Be safe!