The kernel-header error is misleading - you have to install kernel-source. (After you do that, you might have to update the kernel itself, so check that afterwards.)
Yes. If you strip them down (compile/make modules only things that you need), compress the kernel (optimize for size)... it'll result in such a small kernel. You can make an even smaller ones, for targeted system. Like I know my PC, I know my stuff. So I'll just compile what I need. Like my own SATA drivers, driver for USB, etc. Nothing else. No webcam, no old IDE drives, nothing.
Also: The bigger ones you encounter are the WHOLE kernel (which one never needs to use). Even the desktop ones contains (most of the times, as modules) a LOT OF unneeded stuff. But newbies expect that every kind of hardware they plug-in will just work.
(Note that, the modules won't get loaded until they are needed. There are 3 states:
[ ] - Won't get compiled, won't be part of your kernel.
[M] - Will be compiled as a module.
(You can modprobe it, or your system will auto-load it when it's needed.
Depends on the system, userland.)
[X] - It'll be compiled into the kernel AND will be present - always.
Distros with small kernel:
Best Answer
There no universal standard, but the kernel is usually found in the
/bootdirectory.