The fire doesn't rise because your objects are much bigger than in the video (~7 m sphere vs ~1.3 m sphere). Compare the size of the turbulence force field (2 m) with your sphere's size.
The fix is easy:
- scale down the sphere to about 1.3 m and apply the scale to the sphere.
- scale down the domain. Make it only as big as needed. Apply the scale to the object (Ctrl+A in Object mode). The guy in the video didn't do it, but it's good practice and saves your nerves as this can cause side effects and problems when used with the physics engine.
- disable the [_] Dissolve in the Domain setting
Also, in the video the contrast of the texture is set to 5, yours is still 1. I'm not sure about the effect. But double-check it.
You have set Reaction Speed to 0.2. Because of this, the flames are now a bit higher than in the video. After the correction of the size, the fire will look nice:
About The Shader
You already started to create the node setup for the shader. The flames still look different because you have not yet set the Blackbody Intensity value and Density values of the Principled Volume node to
Also, check your node setup and make sure you have only one Material Output node (scroll to the right to find the 2nd one).
(Tip: If you render on GPU you might want to increase the Render Settings > Performance > Tiles to 256. The GPU works often faster with greater tiles, while the CPU is faster with smaller tiles.)