To answer your question about why the update is not installing first, the software is probably downloading the wrong BIOS update file. To update it, I would go to the Asus web site, look up your model, and download the update from there, then follow whatever instructions it provides to install it.
That said, BIOS updates can be a risky procedure. I've never heard of one going wrong, but if your laptop was to lose power during the update, it would theoretically be damaged beyond repair. So the general rule of thumb is not to update your BIOS unless there's a reason to. The only reason I can see for you to update yours is to take advantage of any possible optimizations made in the new version, which might possibly provide some kind of performance benefit (it's a stretch, but it's all I can think of).
If you decide not to update, and wish to be rid of the notifications, I would look around for any sort of Asus settings panel that might contain an option to turn them off.
If that fails, you can download Spybot (from http://www.safer-networking.org) and turn on Advanced Mode. You can then go to Tools (bottom left-hand corner) and then System Startup. You will be presented with a list of every program that runs when you start up your computer. Somewhere on that list is the piece of software that is nagging you about your BIOS. I would just untick the check boxes of all of the entries that mention Asus. Usually manufacturer's software is just useless anyways.
Good luck!