Is there a default parameter choice for the spike-and-slab prior? In the spike-and-slab prior, one needs to specify $h_{0j} = P(\beta_j=0)$, which demonstrates our prior belief about how likely $\beta_j$ to be an important predictor.
Is there a default choice for this $h_{0j}$? Perhaps one that leads to a prior uniform distribution over the entire model space?
 A: This is actually described in the paper you quote:

So it is an locally uniform distribution that defines the same probability of all the values with a "spike" (i.e. larger probability) for some value. Let me change the notation for a moment, say we have spike-and-slab distribution so that $x \in [-\alpha, \alpha]$ with a spike at point $\beta$. If the distribution was uniform then probability (density)  of some $x \in [-\alpha, \alpha]$ would be $\frac{1}{2\alpha}$. Knowing that you can choose some value for your parameter ($\gamma \in [0, 1)$) that reflects your prior assumptions about the model, i.e. how much more probable you would like the $x=\beta$ value to be comparing to the other values. Then your pdf becomes:
$$
  f(x)=\begin{cases}
  \frac{1-\gamma}{2\alpha} & \mathrm{for}\ -\alpha \le x \le \alpha\ \mathrm{and}\ x \neq \beta, \\
  \gamma & \mathrm{for}\ x = \beta, \\
  0 & \mathrm{for}\ x<-\alpha\ \mathrm{or}\ x>\alpha.
  \end{cases} 
$$
In this case the bigger $\gamma$ is, the smaller is probability of other values, so choosing different values will make $x=\beta$ more or less probable comparing to other values. It is your subjective choice to decide how much more probable you want this value to be.
In general, there is no such a thing as "default priors". Priors are always problem-specific and reflect your prior assumptions. You can use "uniform everywhere" approach but this also reflects your assumptions about uniform distribution of your parameters. 
