Hi, and welcome
for the cube, you should multiplex your leds and use shift registers and transistors. that way you don't need to get all the pins from the arduino.
a good instructable about shift registers can be found here
for the alarm clock, you'll need an RTC (real time clock)
'lcd screen' is extremely vague. do you want a 16x2 lcd screen, or a 20x2, or a 16x4, or a full color lcd, or a nokia rip-of-lcd or.. the possibilities are endless.
an on/off button can be a switch between the power source and the arduino, that is the most simple way to turn it off.
the other products are standard and info can be found on a million pages in the internet