which distance sensor should I use for a project like this?

Minimum distance to be measured is 20cm, max is 2, ideally 3 meters.
Need to measure a distance from a white, curved, not shiny wall to a pico video projector and use the distance to control the projector's focusing lens knob with a servo motor.

What distance sensor should I use? there are IR based, laser based and ultrasonic based ones. Not sure which one best suits this project. Just need the distance from the projector to a point straight in front of it.

Ultrasound may work. The popular and dirt cheap HC-SR04 does 2-300 cm, and gets some 2 cm accuracy. If that accuracy is not good enough, you may have to look at optical solutions which cost quite a bit more but are more accurate, too.

2cm accuracy is plenty for projector lens focusing.

The projection surface that needs to be measured from is probably going to be made from white lycra / spandex.

If these are not game stoppers for ultrasound sensors seem like a good option from what you described.

Are there better/expensiver versions of the HC-SR04? For prototype it very likely will get the job done but I wouldn't mind more distance for some $20 more cost.

That'd be a rather smooth surface, right? Doesn't sound like a show stopper to me. Most surfaces work fine.

There exist ultrasound sensors that do up to 10 meters - do a search at this forum, it's been discussed before. I don't have the specific type numbers at hand.