Before I reinvent several wheels all over again, can anyone point me to a solution that would allow me to edit data tables on a PC (e.g. a 16 row by 16 column table) and allow me to download that data into a running Arduino sketch?
I'm not aware of a ready-made solution for that and I think it would be necessary for your sketch to provide an interface to allow the data to be read and written on command from the PC. The sketch would then supply and receive the data and store it wherever you're keeping it (perhaps EEPROM if you want this to be non-volatile).
Note that the EEPROM isn't very big so you will need to be careful what word size you use for your table.
Besides transferring the data over some serial connection (USB, bluetooth, ethernet, wifi, etc.) from the PC to the Arduino, a second possibility is the old sneaker-net approach. Add a microSD card to your Arduino. Use a card reader on the PC, write the data, move the card over to the Arduino, and then use a button or something to tell the Arduino to read from the SD card.
A third approach is to use EEPROM if your Arduino supports it. EEPROM is not reset when the Arduino is rebooted. So you would download a sketch to the Arduino that either reads the table from the PC or it just writes the table directly to EEPROM (you would edit an include file on the PC with the data table). Then you reflash the Arduino with the program you want to use, and it can read the data from EEPROM.
Obviously, depending on the size of the data, you will have to worry about whether you have enough SRAM to hold it.