Is it possible to update the Linux kernel without installing a new version of the Linux distro? If I want to download the kernel from the Linux website and manually install by compiling it, how can it be exactly done?
Yes you can update linux kernel without updating your linux distro.
To install the latest kernel you can always visit http://kernel.ubuntu.com and https://wiki.ubuntu.com/Kernel to see what's going on. As of today the latest kernel is 3.17.
If you are using ubuntu 14.04, for example goto this link: Install/Upgrade to Linux Kernel 3.17 in Ubuntu 14.04
Similar tutorials are for other Ubuntu version and other linux kernel version are available, just google it or comment here i'll help you.