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A lot of DOS programs using 32-bit CPUs would embed a "DOS extender" which would take care of the switch to 32-bit mode and needed translation of interrupts and device accesses, forming another layer of mini-OS on top of DOS while leaving the "launch application from DOS" flow unchanged to the user. Windows 3.1 also had Win32s which provided partial compatibility for 32-bit Windows applications, though that wasn't directly related.

Yes, here is some info,

https://en.wikipedia.org/wiki/DOS_Protected_Mode_Interface

https://en.wikipedia.org/wiki/DOS/4G

Yup. One of the first 32-bit programs many people owned was Doom.

The defacto support, such as it was, was "run DOS/4GW".

An actual MS-DOS running in protected mode would be an interesting thing. Win95/98 might qualify(I don't know enough details, but it did identify as DOS versions 7/8, and they dropped support for pre-386 CPUs).