What are the assumptions of an ordered probit model that must be met? What are the tests to check these?
There are some distributional assumptions about the error, but these cannot be tested in a formal way (as far as I know).
There is also a parallel regression assumption, which is frequently violated. Long and Freese's categorical dependent variables book describes an approximate LR and a Wald (aka Brant) test (and provides Stata code).
The error term needs to be normally distributed to justify the ML estimation of the model. This assumption can be tested using the LM test developed by Johnson (1996) "A Test of the Normality Assumption in the Ordered Probit Model," Metron, LIV, 213-221. The issue is further discussed by Giles at https://davegiles.blogspot.com/2015/06/specification-testing-in-ordered-probit.html.