While a brownout/voltage sag could possibly cause something like this, I think it’s more likely that it’s just simple electrical noise from the microwave that is scrambling the Keurig’s brains. Microcontrollers in general tend to be fairly robust, but the ones that they put into consumer items like a Keurig are the el-cheapo crap controllers, and I doubt that the designers would add much circuitry (if any) to protect the microcontroller from noise spikes.
Either way, relocating the Keurig to a different circuit should eliminate the problem.
It is pretty dangerous. The way that most homes in the U.S. are wired is that they are fed from what is called a “split phase” transformer, or a single transformer with a center tap. You have 240 volts AC from one end of the transformer to the other (line to line, in electrical terms), and 120 volts from either end to the center tap. The center tap is the neutral, which gets grounded at the service entrance to your home.
If the neutral connection develops a problem, then the neutral wires inside the house tend to “float”. You still have 240 volts from line to line, but the neutral is no longer at the zero point between them. It becomes a voltage divider, and the voltage depends on the load connected to the two lines. Basically, roughly half of your house should be connected to one line and the other half of your house should be connected to the other line, with all of your 240 volt appliances (oven, dryer, etc) connected from one line to the other. The loads on each line won’t be perfectly balanced though, so as one half gets dragged down, the other goes up, so that the total still equals 240. In other words, if one line to neutral gets dragged down to 100 volts, then the other line to neutral will measure 140 volts, so that the total is 240 volts. The neutral is “floating” between the two line voltages.
When the voltage goes low on one line and high on the other, this can damage electronic devices and, especially on the high side, it can cause excessive currents which can start fires. In other words, this is the type of problem that can burn your house down and kill everyone inside. So yes, it is indeed quite dangerous.
It’s possible that something like that is happening here, but you would notice it in other parts of the house, namely that some lights will dim and others will get brighter as things like the microwave or vacuum cleaner are turned on and off.