What are the skills that can be done while moving?

Leap slam and vortex are done while walking and do not cause stalling. Are there any other skills that do not disrupt the movement apart from these two skills?

Cyclone, Righteous Fire, Totem/Ballista stuff with enough placement speed doesnt really stall (leapslam without attackspeed does btw)... sure i miss something

Any skill that have the instant "no casting time"

https://pathofexile.fandom.com/wiki/Instant

Mostly Aura/Buff/Guard skill.
Some Movement skill "Like blink"

Most movement skill does have a casting/attack time but the first cast is faster.
If you spam them then it slow down on the second cast/attack.

Every skill that has zero castime. Guard skills for example. You can put them on left mousebutton and never have to care about them. If you are in for offensive skills, well cyclone dont interrupt your movement and goes fluid into spinning motion in and out. The first cast of flame dash is also instant even if the gem says it has a castime. The first cast is still always instant. But in the end there are not much skills that you can use while moving.

Totem and ballista stop movement during application. While I was walking with the mouse, I asked for a skill that did not disturb the movement when I applied it from the keyboard. cyclone and rf yes in this category.

I think i get what you want. Google for poe Autobomber builds. They dont stop at all.. Just moving around and boom boom everything on screen. They are not cheap though.

Flicker Strike and Consecrated Path teleports you around while attacking but not technically moving.