"A new kernel" ???
What is wrong with the old one?
As above building your own from sources like possible Gentoo or Linux from scratch will provide the most exact kernel for your situation.
Distro's tend to be good for many people. Distro's offer a way to keep packages and dependency's closely matched. A kernel will not simply allow you to have the latest whatever by itself.
Your question almost can't be answered. Both yes and no may be correct. We'd need to know much more about this to be sure. It could be that a generic kernel could be swapped and work.
"There are two kinds of kernels in Slackware -- the huge kernels, which
contain support for just about every driver in the Linux kernel. These are
primarily intended to be used for installation, but there's no real reason
that you couldn't continue to run them after you have installed. The
other type of kernel is the generic kernel, in which nearly every driver
is built as a module. To use a generic kernel you'll need to build an
initrd to load your filesystem module and possibly your drive controller
or other drivers needed at boot time, configure LILO to load the initrd at
boot, and reinstall LILO. See the docs in /boot after installing for more
information. Slackware's Linux kernels come in both SMP and non-SMP types
now. The SMP kernel supports multiple processors, multi-core CPUs,
HyperThreading, and about every other optimization available. In our own
testing this kernel has proven to be fast, stable, and reliable. We
recommend using the SMP kernel even on single processor machines if it
will run on them."