does the shoegazing sound depend on feedback?

Feedback doesn't really play much of a part in that kind of music. It's all about effects such as reverb, delay and tremolo. Preferably with at least two or three of them chained together to create the wall of sound type effect.

I imagine you would still need some kind of amplification at the end of that chain though. Never tried it so I could be wrong, but I don't think a guitar going into effects going into headphones would be enough to produce much of a sound at all. Guitar --> Effects --> Amp --> Headphones would definitely work. Just be careful with the volume.

In short, no. The only time you generally stand close enough to the amp to get feedback is just as a special effect and this is only done rarely. You can't play an entire song or even lead with feedback. A common use is during a lead while a note is ringing out, you will stand close to the amp to get feedback for just a few seconds and then step away. Yes effects are how you would accomplish the tone. I am not an expert on the genre, but in general I think they just use distortion and a good bit of modulation, whether it be flanger, phaser, or whatever. Also a good delay pedal.