That keypad will not work for you unless you can figure out how to power it. While its external power port for readers is 5v, the device itself runs at 24v and appears to draw as much as 600mA, which is pretty hefty. You'd need a large, heavy battery to keep it lit and running for very long. That said, it's also not documented how to run it with an Arduino, which can make it really hard for a novice.
Your biggest issue here, as a novice, is going to be managing how many pins you'll need in order to make this all happen. Using an UNO, you're very limited on the number of I/O pins. You can cheat and get a MEGA if you want to go cheap on almost everything else below. Power / battery / driving a large horn+strobe isn't going to be DEAD EASY, but it will be easier than some of this:
1. In order to drive as many digits as you want, you will either need to go with a serial display (two, actually... See below) or use a rather large number of Arduino digital pins to drive the displays. If you choose to go the cheaper route, you will need to use shift registers which, although not all that hard to use, are not trivial for many novices.
2. Your keypad will likely take up 7 pins on your arduino. Given that the UNO (the replacement for the Duemilanove you mention) has only 11 dedicated digital pins, assuming you keep the serial and analog pins for serial and analog uses, this will kick your butt a bit. Again, shift registers can be used, but NOT easily in this case as there is very little out there in the way of documentation for this scenario. http://www.arduino.cc/cgi-bin/yabb2/YaBB.pl?num=1258567421
is just about the only example I've seen that is likely to work and involves extra work.
I link to a serial version of the keypad below. It's worth noting that if you want to use a serial keypad AND the serial displays in #1, you will need to use SPI to communicate with them, or setup SOFT SERIAL (it's a library out there) to use different pins than the dedicated serial pins.
3. MORE PINS! You will need either another 6 pins OR a shift register setup OR (my recommendation) another soft serial connection to drive your LCD. You can find a nice serial LCD in my links below. Writing to it is easy as pie and there are tons of tutorials out there.
You might also try:http://www.adafruit.com/index.php?main_page=product_info&cPath=37&products_id=347 This one is a VFD instead of an LCD, but it's different from the other videos, so that might have a cool factor. Also: Adafruit products almost universally come with beginner level tutorials that can really help!
4. about that buzzer / alarm? Your lowest stress bet is going to be a high volume piezoelectric buzzer. I've linked to one below. You will NOT be able to drive this directly from the Arduino's digital pins. You will need to switch a transistor or mosfet to drive it from a higher voltage external power source. Step up converters will likely not be able to provide the current it needs, so having your battery be 10-12v would be helpful. Just remember you'll THEN need to use a stepdown regulator to drive the Arduino itself. Another option would be to simply use two batteries.
As an aside, you might also look into using ONE LCD to display everything from countdown to input. For this, you could use either a straight 16x2 or use one of these:http://www.sparkfun.com/products/8884
If you use a GRAPHIC LCD like one of these, you would be able to display bitmaps ala team logo or explosions or other fun.
Must get back to my actual job now, but I hope it works out for you! Remember, boys and girls, never use 3.3v devices connected straight to an Arduino's 5v pins. This doesn't just mean the POWER connection, but also the logic pins. Unhappiness will ensue.
I'm mostly going with Sparkfun as a supplier simply because their site has just about everything you need. Searching will likely find cheaper options, though perhaps not as high quality.
Serial numeric LED display that can be placed side to side (to enable as many digits as you specified):http://www.sparkfun.com/products/9766
One of the keypads shown in a video you linked to:http://www.sparkfun.com/products/8653
(I drew up the datasheet for this one - at least the mechanical datasheet.
Essentially the SAME keypad, but using a serial connection, which would make your life easier - FAR fewer pins used (essentially 1):http://www.electrojoystick.com/proddetail.php?prod=EK005
Serial LCD (note, this is the 5v version. do NOT accidentally get a 3.3v version. It will die horribly before too long):http://www.sparkfun.com/products/9394
The alarm buzzer:http://www.radioshack.com/product/index.jsp?productId=2062396
A circuit that WILL work for driving THIS buzzer: http://www.arduino.cc/cgi-bin/yabb2/YaBB.pl?num=1293472286
If you want max volume, make sure you figure out which power source you're driving it from and use an appropriate resistor to lower the current to the stated maximum 120mA. You can google "led resistor calculator" to get a bit of free advice on the resistor once you've chosen a battery.