What are the key assumptions of ordinal logistic regression and what tests to run on STATA about them? As the questions specifies, I am looking for the main assumptions of the ordinal logistic regression and what tests should I run in stata to make sure they are okay

Don't run tests to determine adequacy of model fit.  Instead ask yourself (1) is the lack of fit consequential and (2) if I do anything about the lack of fit, do I make things better or do I make them worse?  For (1) there are various model performance measures (e.g., pseudo $R^2$) to use, and for (2) you can use AIC to tell you if a model that doesn't make the assumptions is likely to predict future observations better than one that makes the assumptions.  These approaches are described in detail here and in other resources given here.  The first link shows how the R rms package impactpo can be used to quantify the problem from various different angles.  In short, emphasize the impact of assumptions, not just testing assumptions.
