Why does this 7-segment display not light up?
According to the datasheet for a TI family of BCD-to-seven-segment decoders (http://www.ti.com/lit/ds/symlink/sn5447a.pdf), these control lines are active-low, and should be either left floating or pulled high when not in use (footnotes on p. 3 and 4).

As for the LEDs not lighting up, what input combination have you tried on the decoder? You've left them disconnected on the picture and for all high inputs (=> 15) it outputs a blank display (p. 3)...

(Edit) I see you found that only a few segments light up when you pull pin 3 to ground. That's the Lamp Test Pin, and it should make all the segments light up. If they don't, something is broken. Have you tested all the segments by hand?