Title says it all - does the shape of a wave have any effect on how we perceive it as a sound?
Yes, but not directly - instead, we perceive the different amounts of energy in different harmonics. :)
A square wave only has odd harmonics (the base frequency, 3x the base frequency, 5x, 7x, etc.) and that gives it a distinct character. This is similar to the triangle wave (which also only has odd harmonics, but in different amounts). They are both very tonally different to the sawtooth wave, which contains both odd and even harmonics.
However, since it's mostly determined the relative amplitude of the harmonics (we aren't very good at hearing phase, except between our two ears) it's possible for two waveforms to look different but sound basically identical, if their harmonics have the same energy.
Here is a video which demonstrates that: at 0:02 it plays a conventional sawtooth waveform, but at 0:12 it plays a different-looking waveform which has the same energy in its harmonics, so sounds the same.
Edit: as @MacTuesday mentioned, the distinction between a continuous tone and a quickly-repeating sound can be ambiguous, particularly for very low pitches. Phase/timing information is not meaningless, but for the cases you're thinking of, harmonic energy is the primary aspect that determines timbre of continuous sounds.
Of course. You can trivially prove this with programmable synthesizers or software that allows you to customize an audio wave form for output. There are distinctive differences between regular sine/cosine forms, rectangular waves, and triangular (saw) waves.
You've got it the wrong way round. Waveform is a pictorial representation of sound.
Duty cycle of the waves can also affect the sound. Sometimes just in the perceived volume, sometimes in the tone(s) heard.