1.) EDIT: Am I supposed to supply pin 3 with 1/2 VDD (2.5V)?
Yes two equal resistors (say 10K) from the input to each rail should do it.
2.) If I want the lowest reading from the photodetector to be 0V even though it receives light, how do I do this? Do I make the TLC272 output the differential voltage, and if so, how?
Replace the two 10K resistors with a pot. then you can trim the input to what ever level output you like. You will need to do this because you have a high gain and a DC input.
3.) Will I still need to use signal diodes in this design to limit the voltage?
No because this can't give any more than 5V.