1: Is there a way to emit particles in a linear and structured order like shown in the picture?
2: Even better: Is there a way to animate the particles from random to a structured order seamlessly?
i would recommend to use geometry nodes for that task. Then you have much more control and flexibility.
Here is an example for this:
Node tree:
3D Grid:
you get the animation just by changing/keyframing the factor of the attribute mix node.
Add a particle system to your object, emitting from Volume, rather than faces.
Under distribution change from Jittered to Grid. You can then animate the Random parameter to have it go from unordered distribution to an orthogonal grid.