What are good practices in reporting RMSE or MAPE estimates for a machine learning model?
In general you should not attemt to make your model look better than it actually is. Always stick to the truth. Some ideas:


Outliers: It is not okay to simply drop the outliers. If you want to make a prediction or do inference, you have to ask yourself how relevant such outliers are in practice. Where do they come from (are data reliable or is it an error)? If they are part of the true bigger picture, you cheat if you just drop them. If they are a very special part of the market with specific attributes, you could try to control for these special types (e.g. by adding indicators or changing the parametric form of your model). This question essentially is data and method driven.
Your metrics should of course reflect the final model!
By definition of the RMSE it is the mean (not median). Naturally, you can describe the situation by discussing the influence of the outliers in your own way. But this will not make your model/fit better after all. 


Finally: I think you should NOT ask about a metric that suits your approach and make it look better. You should ask yourself if you can improve your model. You did not specify what type of method you use and what data you work with, but I'm quite sure there are ways to improve the fit. The scatter plot looks like an L-shaped function. So this might be the first thing to start to work on!?