Build kernel module on locked bootloader

To shortcut your two questions: you can't load a module you've compiled yourself without unlocking the bootloader.

Technically you can compile a kernel module without compiling the kernel yourself, if you have the kernel sources and you know what options were used to compile the kernel; it's just easier to compile the kernel and modules at the same time. But that doesn't matter anyway, because you need to be able to sign the kernel and/or sign a system image containing your kernel module, in order for that module to load. You need to unlock the bootloader to avoid the signature check.