Addition of step disturbance to a microgrid state space model
The general form of state-space is

\${\dot {\mathbf {x} }}(t)=A~\mathbf {x} (t)+B~\mathbf {u} (t)\$

\$\mathbf {y} (t)=C~\mathbf {x} (t)+D~\mathbf {u} (t)\$

where A,B,C and D represent the state, input, output and feed-through matrices, respectively.

You should keep in mind that these matrices are the same for the same system parameters. Any change you want to impose to the system should be done through input vector (\$u\$), and the output will be observed from \$y\$ vector.

Regarding your first question, from Fig. 2 and equation (64) in the mentioned paper. It can be observed that the load stat-space model has input vector as \$\Delta v_{bDQ}\$ and \$\Delta \omega\$. Therefore, if you want to implement a step change in power, you have to apply step change in one or both of these vector inputs, and as a sequence of that the output vector (load current in this case as per eq. (64)) will change.

Regarding your second question, the results in page 622 of the mentioned paper are not simulation results. It is a study of the impact of changing some control gains and parameters on the system stability. You can simply do it by plotting the pole-zero map of the system/subsystem you are interested in for different variation in control parameter (in Fig. 12 active power droop is the parameter which is changed).