Now that I have investigated the paper in question, here is my new answer.
It turns out that the paper actually discusses task-based motion generation. That is, given a task (end-effector velocity trajectory) $\dot{r}(t)$, find a robot input (joint velocity trajectory) $\dot{q}(t)$ that realizes the task. The formulated optimization problem is happening at a time instant on the trajectory.
So if the Jacobian is square and full-rank, you wouldn't have to worry about this optimization problem since there can be only one $\dot{q}$ corresponding to a given $\dot{r}$. However, when the robot is redundant, there can be infinitely many $\dot{q}$ satisfying $J\dot{q} = \dot{r}$. Now that you have choices, you can do fancier things.
In this case, the minimization objective is $\Vert \dot{q}^TQ\dot{q} \Vert$ to minimize the magnitude of the joint velocity. The reason is simple: one wouldn't want the joint velocity to just go unreasonably high.
So, to really answer your questions
Why is the IK formulation like that?
I think it is just that this formulation is mentioned as an example that leads to the main content of the paper. IK problems don't have to be formulated this way. There are other formulations out there.
How is that giving IK solution?
$\dot{r}$ implicitly specifies the next end-effector pose. A solution $\dot{q}$ implicitly specifies the next IK solution to reach the specified pose.
isn't the optimum value be dq/dt=0?
Without the equality constraint, yes. However, there is this inequality constraint. So $\dot{q}$ can be non-zero.
(Below is my previous answer)
The formulation is general, i.e., it can be used with any types of robots.
I think it may be easier to consider the discrete-time analogy of the problem.
Given an initial point $q$ (i.e., a robot configuration), this optimization problem is actually trying to pull $q$ towards some point $q^\ast$ that is an IK solution to your problem. What is effectively steering the point $q$ towards a solution is the difference $\Delta r$ between your desired end-effector position and your current end-effector position. However, what you control is actually the joint values. They are related by
$$J\Delta q = \Delta r.$$
You can then compute the joint velocity that satisfies the above equation (given the current $\Delta r$). And since $\Delta q$ is essentially the difference of joint values between consecutive time steps, $\Delta q = q[t + \Delta t] - q[t]$, you will obtain a joint value for the next time step.
That's why this is the constraint that is optimization problem is subject to.
At the end, the system should stop moving when it reaches an IK solution. This means $\Delta q$ should be zero. That's why the minimization objective is to minimize a norm of $\dot{q}$.
I think the positive semi-definite matrix $Q$ in the objective function will control how each of the joint values converges to an IK solution. For example, if $Q$ is an identity matrix, it means you place equal importance to each DOF (that is, you do not have any bias in favor of some DOFs to converge faster than the others).