Is there a method to extrude faces randomly to get uneven height of the extruded mesh?
It's a few steps but I found a way do do this.
I Select all the faces you want to extrude and extrude them with no movement. You can also instead use an inset with size 0 if you want your edges to separate the heights.
(if your faces are not aligned with a global or local axis, now would be the time to create a custom axis by clicking the little + in the transform orientations dropdown)
II Use the Mesh>Transform>Randomize Operator with an Amount of your choosing, Uniform set to 0 and Normal set to 1.
III Set your Pivot Point to Individual Origins and scale along the relevant Axis by 0 (the one normal to your faces)
What got me what I wanted was enabling the Add-on Mesh: Edit Mesh Tools from the preferences and using "Multi Extrude".