Do I need fog or haze to make the laser beams visible?

You always need a surface to reflect the laser light. This can be fog or haze for beam shows, or all kinds of screens for projections. Very common screen types are waterscreens, gauze screens, rear projection screens or fog screens.

All 3D-like projections with lasers also need kind of a surface to be reflected on!

It is not possible to create laser-sword like effects: You cannot "terminate" the laser beam somewhere, it can physically not be terminated without having an end-point (surface).

Laserworld recommends to use a haze machine for indoor beam shows to get the smoothest effect.

It is definitely not possible to see any laser beams in the air without having particles in the air!