bivariate probit with endogenous covariate testing

I am interested in learning more about testing for the bivariate probit model with an endogenous treatment regressor. I have figured some stuff out -- summary below, since I don't see much on this topic -- but other questions remain.

Here's the setup. Suppose I have a binary outcome $$y_1$$, which depends on $$x_1$$, the error $$\varepsilon_1$$ and binary treatment indicator $$y_2$$, which itself depends on $$x_2$$, and $$z$$ and error $$\varepsilon_2$$. The dependence is of the form $$y_1=\mathbb{1}(\beta_1'x_1+\alpha y_2+\varepsilon_1>0)$$ and $$y_2=\mathbb{1}(\beta_2'x_2+\gamma z+\varepsilon_2>0).$$ The errors int the two equations are correlated with $$\rho$$.

I am interesting in testing the assumptions of:

1. normality of the two errors terms
2. $$\rho=0$$
3. homoskedasticity of the errors
4. exogeneity and weak instruments for the treatment equation

Normality can be tested with goodness-of-fit Rao/Murphy score test described by Chiburis et al. (2011) and Chiburis (2010), who provide Stata code do so (scoregof). This test embeds the bivariate normal distribution within a larger family of distributions by adding more parameters to the model and checks whether the additional parameters are all zeros using the score for the additional parameters at the biprobit estimate. It rejects when there is excess kurtosis or skewness in the error distributions. They do not recommend a variant of the Hosmer-Lemeshow test to do this based on simulations.

The correlation between the errors can be tested using a likelihood ratio test based on the idea that if $$\rho=0$$, the log-likelihood for the bivariate probit will be equal to the sum of the log-likelihoods from the 2 univariate probits. If you calculate Huber/White sandwich errors, this becomes a Wald test.

Some questions remain.

1. Should I worry about heteroskedasticity if I use robust errors? How can I test for this?
2. Can I differentiate heteroskedasticity from heterogeneous treatments effects?

3. Can I use linear IV diagnostics (like weak instruments tests) to check the non-linear probit model? Is there anything better? This seems strange to me since they estimate different treatment effects.

4. Are there other things I should check that I am not aware of?