Can the kernel update itself via the update manager?

There is a system called ksplice which allows kernel upgrades without restarting, but Ubuntu isn't using it yet and it's a system most attractive to server admins at the moment.

Any time you have a new kernel update which is made available in the software center updates, you will need to restart the computer in order to make use of the fixed security or other bugs. Each kernel version is separate, so keeping your machine running on the existing kernel won't be harmful unless you count potentially security holes. See also this existing question:

Why not include something like Ksplice so there is no need to reboot after updating?

eg. Could I still be running "Lucid" in 2020 with each "latest" kernel being installed via updates. (I'm not going to do that; it's just a test scenario.)

The name "lucid" is really just the name of the distribution. Yes, in theory (and to some extent in practice) you can keep "lucid" till 2020, its just that you will be upgrading all the components in your system.

Upgrading just the kernel itself (and nothing else) works for sometime, but when there is a major kernel change it will be necessary to change / upgrade other programs also or they wont work anymore.

In-place kernel updates were discussed in this question: Why not include something like Ksplice so there is no need to reboot after updating?

As for the reminder to reboot, see the /etc/motd file for servers when you log in:

Linux terra 2.6.35-22-generic #34-Ubuntu SMP Sun Oct 10 09:24:00 UTC 2010 i686 GNU/Linux
Ubuntu 10.10

Welcome to Ubuntu!
 * Documentation:  https://help.ubuntu.com/

*** System restart required ***
Last login: Tue Nov  2 17:33:22 2010 from ...

and the panel notifier reminder for desktop: