Other..Prohibited Items

In the uPD78P4038, setting 0000 to the prescaler mode register is prohibited.
Does this mean that it is not OK to set 0000 to this register?

Yes, that is correct.

When not using a function, I set a setting prohibited value to the related registers. Is this OK?
Does "setting prohibited" have the same meaning in other CPUs too?

This interpretation of the meaning of "setting prohibited" is incorrect. A setting prohibited value is a value that absolutely never must be set.
If such a value was set, it would be completely impossible to guarantee the operation that would be performed.
In the worst case, the CPU might completely stop functioning.

This applies to all registers and all CPUs.
When not using a register, set an allowed value that does not cause the system problems to that register.

The internal peripheral registers include bits that are fixed to 0 or 1. Can a value other than the specified value be set to these bits?
If not, for what reason?

The answer is no. If a value other than the specified one is set to an internal peripheral register (SFR), normal operation may not be possible.
This is because, in addition to the disclosed functions (which have been tested and whose operation is guaranteed), the internal peripheral functions include also undisclosed functions (which have not been tested and whose operation is not guaranteed). Moreover, in some cases, some functions are included for device testing purposes, and if values other than specified ones are set to these registers, malfunction may result.
Basically, the operation of a microcontroller is guaranteed only under the conditions specified in the manual for that microcontroller. Therefore, settings other than the specified ones must not be done.

In the uPD789116, if the microcontroller enters a program loop when the unused space is filled with FFH, what type of operation is performed?

The instruction code FF is undefined code in the case of the 78K0S.
Regarding the execution of undefined code, it is unknown what type of operation is performed by the CPU.
(The execution of undefined code is not tested, and the operation in this case is not guaranteed in any way.)

Is it all right to read addresses in the SFR area to which registers have not been allocated?

Basically the operation when addresses in an SFR area to which registers have not been allocated are accessed is not guaranteed in any way.
Normally, when an undefined area is read, an undefined result is read out.
If SFRs do not exist at those addresses, write becomes a meaningless operation.
But if test circuits, etc., that are not available to users exist, an inadvertent program loop may occur.
Please do not access addresses other than the user-available SFR addresses, or the subsequent operation will not be guaranteed in any way.