Just doing some quick math to see the current requirements of the LED display. The LED strips I have are made for 12v and have a built in resistor. They are separated into groups of 3 serial connected LEDs with the strip itself all in parallel. There are exposed copper pads on a 5CM pitch, where the strip can be cut. 

When connected to a 12V source the entire red strip pulled roughly 1.43A. The green pulled 1.3A. The strip is labeled as 5M with 60 LED per meter. That's 300 LED total and 100 total 3 LED groups. That gives 14.3mA per LED group. I figure for the 7 segment style display, 2 5cm segments would give 20cm x 10cm digits, which should be pretty visible from a distance. That's two groups per segment for about 28.6mA per segment. That will be the bear minimum required to drive each segment. 

Assuming I allow for 13 digits (time (4), score(green/red)(4), team score(green/red)(4), and period (1)), I can drive the display as a 7 x 13 matrix. This give a potential for 13 Segments on in parallel for the current source, and 7 for the current drain. That give a minimum of 371.8mA per output to drive the display. 

My switches I have chosen look like they can handle these requirements. 

http://www.irf.com/product-info/datasheets/data/irlml2060pbf.pdf