glibc allocator, like pretty much any other allocator, maps pages in advance without touching them. It's quite likely that you'll get OOM killed when you first try to touch them, rather than during mmap() for new pages.
(This is documented in https://www.kernel.org/doc/html/latest/mm/overcommit-account... )
That's in kernel-space, not user-space processes. If you have overcommit disabled, it will be user-space processes making syscalls mmap() or sbrk() to get more heap memory (in which small allocations reside), that fails with an error return code, I assume. But I guess it could also fail to auto-grow the main thread stack, and presumably get a segfault. If the kernel needs to allocate more memory for a fork(), that syscall can fail. But if the kernel needs to allocate memory for some other internal purpose, probably back to the OOM killer, I guess ... hopefully this would be very rare?
Nope. No overcommit simply means that the OOM killer will be invoked earlier. The errors likely won't happen on the synchronous path due to the way glibc() maps the pages.
Long ago, I tried to write an OOM-resilient container library, and I wanted to test it. It worked as expected on Windows, but on Linux I was not able to get NULLs from malloc(). No amount of MLOCKALL and other tricks helped.
By killing your process and/or by failing fork()s.
In general, malloc() in Linux will never fail for small allocations.