Feed the horse and Tricks will appear. Keep training the horse until the tricks are mastered (it will lower the horse's energy each time and you will need to keep feeding it).
The you can Take the horse and it will shorten your travel time. There is also at least one task that requires you have a mount with you.
Use the Feed button. It's in the screenshot. The horse eats carrots if I remember correct. You can buy carrots from the shopkeeper Bernard. But there's no harm in just clicking it first and see what happens.
If you don't see the same menu I posted, click the tools icon at Home, then the horse icon (Animals). The menu you described (Like, Dominance) I have only ever seen in the Prison Cells, and you can't keep a horse in prison, so I am not sure what it is you are looking at.
The only other way I know to interact with your animals is at home, 'Check outside " then "Check animals", but the only option for the horse there is "Name horse".
