Simulate and compare the responses of a linear and non linear system
How do I find the equilibrium points and write the dynamics of the system in state space form?

You should first determine the state space model. Look for all the variables that uniquely characterize the dynamics/evolution of the system, they will be the state. (Hint: look at \$H\$ and how it changes.)
Once you determine the state \$x\$ of the system and its dynamics \$ \dot{x} = f(x,u) \$, where \$u\$ are the inputs of the system. The equilibria \$x^{eq}_i\$ are were \$\dot{x} = f(x^{eq}_i,u^{eq}_i) = 0\$, and you can have many of them. But in your question (homework?) you will be able to find infinitely many equilibria (for any \$H\$ you can find at least one \$u\$ that will make that point an equilibrium).

Also how do [I] linearize the system [?]

Once you figure out \$ \dot{x} = f(x) \$, which will be the nonlinear state-space model. For each equilibrium point you can have a linearization, and its parameters will be
$$
A_i := \frac{\partial}{\partial x} f(x,u)\Big\vert_{x = x^{eq}_i \\ u = u^{eq}_i} \\
B_i := \frac{\partial}{\partial u} f(x,u)\Big\vert_{x = x^{eq}_i \\ u = u^{eq}_i} 
$$
And each resulting linear model will be
$$
\dot{z} = A_i z + B_i v
$$

... compare the responses of linear and non-linear systems in SIMULINK?

You will have to code the whole thing in SIMULINK, one simulation for the \$\dot{z} = A_i z + B_i v\$ linear model and one for the \$ \dot{x} = f(x,u) \$ nonlinear model.