Is it possible to use a system with a modern, multi-core CPU with older operating systems that only support single core operation? Is this something that needs to be set in the BIOS? Does it depend on the CPU, the BIOS, the OS, or something else?
The previous answer talks about various issues that may arise when attempting to run an older OS on a modern system, but doesn’t really focus on the questions asked.
When a multicore CPU powers on, a single core is activated. The BIOS typically has a configuration option to control how many additional cores should be activated. If this BIOS setting is set to a single core, then any OS that boots will see only that one core.
When the BIOS starts up the boot loader, followed by the OS itself, again that is done with only a single core active. The BIOS reports the number of cores available to the OS. If the OS supports multicore operation, the OS activates the additional cores. If not, the OS will proceed on the original single core. Operating systems such as Linux have a boot option to specify the number of cores to activate, so the OS can be limited to one core. This is useful for debugging.
On a processor with hyperthreading, every place I wrote “core” above can be replaced with “hardware thread”.
Preface: I assume the question to be deleted or moved soon, as this is not in any way RC related. After all, the issue in question is not the classic OS, but workings modern hard- and software.
Since every multi core CPU starts up in single core mode, it is basically possible to use any old, single core OS. As long as the CPU is compatible - which no longer always is true - but that's less due having multiple cores (or threads) but changes in basic CPU design.
In compatibility mode even a modern BIOS should be able to to initiate classic upstart procedures. With modern (UEFI) BIOS, you may have to activate CSM mode (Compatibility Support Module).
Past that hurdle it depends much on the OS to be used. Emulation on BIOS level for like Kbd/Mouse and basic Video will work fine. Even basic Disk may work (Size and filesystem may of course be a bigger issue).
As soon as direct hardware drivers are installed, the chances will reduce. While chipsets do emulate several classic ports one way or another, they are not really foolproof. In fact, they are usually only tested to boot some minimal DOS for BIOS updates, not so much for OS support in all possible configurations. So DOS 1.25 may work better than 5.0 with its GUI. Similar systems that rely on special, no longer supported processor modes - like 286 protected mode and alike got a high chance of failure. So I wouldn't really try for OS/2 - not even 1.2 without any GUI.