What type of processor in usual 8-digits pocket calculator?
There are surely a lot of different ICs around for that application.

One I found was EMPCD081A, which has all the features you mention, as you can see in the datasheet, it runs with a 200kHz clock during operation (while calculating I assume) and 18kHz during wait operation. It can't shut down completely as it has to refresh the LCD and register a key press on the matrix. Running this slowly allows it to operate from a solar cell under very dim lights. It just takes 20µA maximum during operation.

My quick search has not revealed anything more specific on the internals of the chip (bit width of the registers).



Doing some further digging, there are some sources which suggest that they internally work with a 4-bit logic:


TC83220-0009: has a 4-bit oriented RAM (runs at 4MHz, but it can drive a printer, so it's a bit different)
MM57109: This is a really old design, but in this file a block diagram is depicted showing a 4-bit bus.