So, what do you mean when you say I don't need to know the status all the time? I planned on having the LCD screen(now leaning towards the matrix as mentioned by promacjoe) act according to the status of each and every reed switch on the board. Is this the wrong way to go about it?
What he means is that you don't need a very rapid polling rate for the reed switches. Something in the range of 100 ms maximum response time will feel instantaneous to any human, so scanning one row of switches every 10-15 ms leaves a huge amount of time left to process other things.
A matrix of MxN elements will use M+N IO pins. The LED and reed switch matrixes would each use 16 pins if connected directly to the microcontroller. You can reduce that pin count by using external chips like port expanders or shift registers.