Common assumptions are that
$$
\textrm{Cov}(\mathbf{u}, \mathbf{e}) = \mathbf{0}
$$
$$
\textrm{Cov}(\mathbf{e}) = \sigma^2_e \mathbf{I}.
$$
Let $i \neq i'$.
On the one hand, we have
$$\begin{align*}
\textrm{Var}(y_{ij})
& = \textrm{Var}(\beta_0 + u_j + e_{ij}) \\
& = \textrm{Var}(u_j + e_{ij}) \\
& = \textrm{Var}(u_j) + \textrm{Var}(e_{ij}) + 2 \textrm{Cov}(u_j, e_{ij})\\
& = \sigma^2_u + \sigma^2_e.
\end{align*}$$
On the other hand, we have
$$\begin{align*}
\textrm{Cov}(y_{ij}, y_{i'j})
& = \textrm{Cov}(\beta_0 + u_j + e_{ij}, \beta_0 + u_j + e_{i'j}) \\
& = \textrm{Cov}(u_j + e_{ij}, u_j + e_{i'j}) \\
& = \textrm{Cov}(u_j, u_j) + \textrm{Cov}(u_j, e_{i'j}) +
\textrm{Cov}(e_{ij}, u_j) + \textrm{Cov}(e_{ij}, e_{i'j}) \\
& = \sigma^2_u.
\end{align*}$$
Hence
$$\begin{align*}
\textrm{Cor}(y_{ij}, y_{i'j})
& = \frac{\textrm{Cov}(y_{ij}, y_{i'j})}{\sqrt{\textrm{Var}(y_{ij})}\sqrt{\textrm{Var}(y_{i'j})}} \\
& = \frac{\sigma^2_u}{\sqrt{\sigma^2_u + \sigma^2_e} \sqrt{\sigma^2_u + \sigma^2_e}} \\
& = \frac{\sigma^2_u}{\sigma^2_u + \sigma^2_e}.
\end{align*}$$
The latter is the correlation between measurement $y_{ij}$ and measurement $y_{i'j}$ ($i \neq i'$), i.e., the correlation between "any two responses having the same $j$".
Concerning the display of the results, specify the option variance if you prefer variances over standard deviations.
Concerning the significance, you can run an OLS of the dependent variable on all independent variables with exception of the level 2 identifier (i.e. schools), using the command regress.
Store the estimates you obtain through estimates store [name1].
Then estimate your multilevel model using xtmixed and again store the estimates by estimates store [name2].
The difference between these models is the random intercept you allowed in the multilevel estimation but not in the OLS estimation; hence testing whether the unconstrained model performs better is equivalent to testing significance of the random intercept.
lrtest [name1] [name2], force will do this for you. You will need to specify the force option; otherwise Stata deems the test invalid.
Best Answer
John B. Nezlek argues that ICC should not be a ground for justifying decisions on multilevel models, because it's values could be misleading. In his article he gives a synthetic example of varying within-group relationships when intraclass correlations are 0 (attached below). So some, like Nezlek, would say that this is not a problem.
See: Nezlek, J.B. (2008). An Introduction to Multilevel Modeling for Social and Personality Psychology. Social and Personality Psychology Compass, 2(2): 842–860.