In terms of controlling a door lock, you might like to consider an electronically controlled door strike (e.g.: 12V) - they can be powered by a 12V supply and could be controlled by a relay activated by the Arduino. You'd need to decide whether you want the lock to fail open (unlocked) or fail closed (locked) in the event of a power failure - this would of course depend on the safety factors for the door in question. If it was an exterior door, you'd want it to fail closed (locked) so that cutting the power does not grant entry to the door, however, you would need to make sure that if anyone was stuck inside, they could override the lock manually to exit the building to avoid safety issues. This would, in my opinion, be easier than trying to get a servo motor to turn a locking mechanism, and not that much more expensive either.
In terms of the button logic, there's plenty of tutorials on how to read a button - e.g.: this one from the Arduino site itself.
Likewise the LCD display.
Not sure about the IR sensor side of things - both of the ones you link above are fairly limited in range (1-3cm for the first and I suspect a similarly short range for the second).