The name you enter should be as close to the name you have registered with British Dressage.

Events are only shown in My Events when your attendance has been confirmed. You can do this by using the Going button for an event but once the event has published times it should be set automatically for you.
NB If the event does not publish online information then you will need to use the Going button.

This is normally caused by a lack of favourite venues. Check you have venues marked as favourites by going to Settings > Venues and confirming the venues you use most often are shown with a filled in heart. If they are not, clicking on the heart will change the venue status and should trigger a check for upcoming events.
If you still don't see any events, then it is possible that there may not be any events at those venues for your current competing levels. e.g. if you only compete at Quest then affiliated events will not be shown.

Woohoo. Progress! To change your levels, select those that apply by going to Settings > Your Name and change the options shown there. As soon as you change the levels the app should search for suitable events and add the Affiliated option to the bottom menu.
You can also change the levels for Friends & Family riders.

Oh. That's not ideal. Is the result listed on British Dressage? They often take a few days to process results so it could just be that it's in the process of being added.
If it is listed on the British Dressage website then there are various reasons the app may have failed to add it and we will need to look at why. Please get in touch with the details of your test and we will investigate.