I need some clarification regarding module building:

When I want to add a kernel module, do I need to:

Configure and compile the kernel using
make kernel_menuconfig
AND build the firmware using:
make menuconfig; make

OR I just need to:
make menuconfig and select the required kmods and then: make

OR I need to opkg install kmod-xxxx (although it is presently complaining for compatibility with the running kernel) to the running firmware?

You have to make menuconfig/kernelconfig - depends on what you need. Then build firmware with make. In some cases you can install kernel modules with opkg install.

I know this is more a OpenWRT question on how things are organised, but it is not clear to me if for building a module I need both to enable the kernel option AND the corresponding kmod package from the OpenWRT menuconfig. One would think since there is module support built into the kernel, all you need is the correctly compiled module, installing the kmod package even on the running system would work. I've also seen cases in openwrt forums where people suggest forced install of kmod packages from different kernel builds, which simply work.

It might work, and might not

Answers probably would be more specifically, if questions were too