The pin numbers in the sketch do not match the actual pin numbers in the processor. For example, the 12th pin on the processor is referred to as 9 in the sketch.
Anyway, I had a lot of problems with the ATTiny, especially with the programming setup, so I decided that I'd switch to ATMega328. At least I shouldn't face any compatibility problems and it isn't that much bigger anyway. I guess the ATTiny could've worked with an external clock, but I'd rather not waste resources and time just to come up with another problem. Using the same processor that the Uno has seems like the best choice for now.
Just in case anyone else is having trouble programming a chip with the arduino, try making your wires shorter than 3 cm (1.18").
I don't know if that's a no-brainer for you guys, but it wasn't for me.
Other than that, I've made some nice progress. I've got the code working, the parts should be available and I've done the schematic and the board with Eagle.
Though I'm gonna have to do the board again since I made the wires a little too small.