On a computer with a single cpu core, must a multithreading program be implemented based on an OS?
No. A program can implement its own threading and scheduler. At this point, however, the program is taking on the role of an operating system, in that in the absence of a proper operating system kernel, the program must implement all of the hardware communications code normally found in drivers.
Can it be implemented in the instruction set and run without an OS?
Yes and no. The instruction set of many modern CPUs includes the functionality required for hardware-level threads, and this functionality must be implemented in the form of a thread scheduler. That may indeed be part of the program, but again, in the absence of an operating system, the program must implement those features that communicate with hardware.
Can multiple programs run in multitasking way without an OS? Can multitasking between programs be implemented without an OS (e.g. implemented only in the instruction set)?
This suggests a confusion about the role of the operating system.
In order for distinct, disconnected, independent programs to execute in a multithreading environment, a process monitor is necessary to manage the scheduling, execution, and termination of these threads. At an extremely high level, the operating system kernel is, at its heart, a process monitor.
On a computer with multiple cpus/cores, must a multithreading program be implemented based on an OS? Can it be implemented in the instruction set and run with parallel threads without an OS?
There must be some piece of code that schedules code execution on multiple cores, as well as to manage the initialization of those cores. This is normally the role of the process monitor. A single, self-contained program could in principle implement these features, but, as above, it must take on all of the roles of the kernel in order to do so.
Can multiple processes run in multiproccessing parallel way without an OS? Can this multiproccessing parallelism be implemented without an OS (e.g. but in the instruction set)?
Again, this is the role of the program monitor. On typical modern general-purpose CPU architectures, such a monitor must exist in order for processes to be scheduled on cores other than the primary core. In principle, a CPU core could be designed to deal with these matters, and I believe some GPUs and certain other small cores (GreenArrays comes to mind) are in fact built this way, or, at the very least, rely on an ancillary general-purpose CPU to run the process monitor.
Are multithreading, multiprocessing,and multitasking, respectively, intrinsic to computer architecture or to operating system?
Are thread and process level parallelisms intrinsic to computer architecture, or to OS? Is multitasking ( as a kind of concurrent computing) intrinsic to computer architecture or OS?
No, and no.
Again, I detect a misunderstanding of the role of operating systems.
There are aspects of parallelism that are implemented in either the computer hardware or the operating system, but they are intrinsic to neither. Historically, this was entirely the domain of the process monitor, especially in early time-sharing systems. That role was placed in the domain of the system libraries in cooperative multitasking systems, where some system calls temporarily hand over execution to the operating system to handle task scheduling, among other things. Modern multiprocessing operating systems are little more than process monitors that manage, among other tasks, processes dedicated to hardware management.