I think the answers to your questions may depend on the particular monastery and where you are in formation.
At the monastery where I am an oblate, there are a couple of dogs. They don't belong to a particular monk, but to everyone. There are no horses though perhaps there are monasteries that do own horses. It seems like it would be pretty expensive to board and care for them unless they fit into the work of the monks.
Yes, the monks go outside. They really have no choice if they want to get to the chapel or refectory. Their days seem pretty full to me so I can't say that I see them simply sitting under trees. They have lots of prayer time along with work time, meal time, study time, etc.
Have you visited a monastery? It would be a good way to get a feel for what their lives are like.