What’s the setting range altimeters can handle? Is there a standard requiring a certain range for aircraft to be certified? Are there many occasions where a QNH is so high or low that airplanes have problems handling that?
There are actually multiple different answers depending on the specific type of altimeter system and what standard applies.
The stand-alone altimeters in older aircraft and small GA aircraft (panel mounted "steam gauge") are certificated to SAE AS392C, Altimeter, Pressure Actuated Sensitive Type. The specified scale range is "at least 28.1 - 30.99 inches of Hg (946 - 1049 millibars)".
This standard was updated in 2016 to require compliance with SAE AS8009C, Pressure Altitude Systems. Altimeters built to this standard have a specified range of 27.50 - 31.50 inches of Hg (931.3 to 1066.7 mb).
Aircraft with "glass cockpits" use Air Data Computers certificated to SAE AS8002A, AIR DATA COMPUTER - MINIMUM PERFORMANCE STANDARD. Barometric corrected altitude correction is covered by Table 2, which defines a range of 22.00 - 30.98 inches of Hg or 745 - 1049 millibars.
The pressure ranges for altimeters in the United States are set forth in a technical standing order. For example TSO-C10b in turn references SAE standard AS392C, which indicates the following performance parameters:
Type I : range to 35,000 feet
Type II: range to 50,000 feet
Both adjustable from 28.1 to 30.99 inHg.
These parameters have regulatory consequences, notably FAR 91.144, which restricts flight operations when the barometric pressure exceeds the upper limit set in the TSO. I have found no regulation for the bottom end, I suppose because the weather would be discouragingly foul.
It is very dependent on the type of altimeter. A lot of standard GA altimeters are rated from -15,000ft to 50,000ft based on a mechanical range rather than a pressure limitation.
For example, on an older model of an aircraft from 1979 a barometer reading of 25.69 was taken from the aircrafts altimeter. At uncorrected station pressure, this would be 71,000ft.
Thus altimeter limits are altitude based, not pressure based*
*In most cases.