Arduino Uno - combined LCD and keypad interfacing problem
You could use a shift register to read the switches as outlined by the excellent tutorial Parallel to Serial Shifting-In with a CD4021BE. Quoting the summary:


  Sometimes you'll end up needing more digital input than the 13 pins on
  your Arduino board can readily handle. Using a parallel to serial
  shift register allows you collect information from 8 or more switches
  while only using 3 of the pins on your Arduino.


So you would only use 3 pins to read the seven switches, which would leave you with 9 pins to play with, for the LCD.

In addition, you could use a similar method for the LCD, using a shift-out register, and use just 3 pins for the 6 inputs of the LCD. A tutorial for that is Alphanumeric LCD with Shift Register on Arduino – part 1, using the AlphaLCD library or 3-Wire Serial LCD using a Shift Register, which has no library, but provides code. 

This would leave you with 6 pins free (out of the total 12 pins, if you exclude pins 0 and 1 for serial), after both the switches and LCD are wired up.

Of course, this solution requires you to purchase one or two shift (in and/or out) registers.

The Arduino IDE may come with a built-in library that use shift registers, but I am not sure about that. Maybe someone can confirm, or refute, this?

I hope that this helps.