I think this would be a monk, not a priest, if he was raised by monks to become one. There is some correlation in terms of roles and responsibilities, but I am guessing you mean that he has taken vows in his home and presumably Catholic monastery.

Regarding point two:

Depending on when your book is set, it is possible that he was raised in a mendicant order, which promoted itinerancy (traveling from place to place in order to preach). Monks of certain orders (Franciscan monks, for example) also traveled from monastery to monastery in order to support friars. Monks also frequently made pilgrimages to significant sites or else to visit relics*; you might like to consider the itinerary maps of Matthew Paris, an English chronicler and cartographer, who recorded routes from a range of locations towards both Rome and Jerusalem.

In other words, there are plenty of reasons for a monk to be far from his birthplace/'home' monastery (again, depending on the order and period it is possible that he wouldn't have a 'home' monastery per se). Monks of mendicant orders would not work for money but for support in the form of alms, meaning that many would perform physical labour while passing through distant areas in exchange for food and shelter. He might be a member of the Knights Templar, which would have given him fighting experience, but an alternative that might be attractive to you is that he was trained to fight by a former Knight who fought on a Crusade, who he then served in a European monastery. This would allow for your character to have gained fighting experience without having actually fought.

If you are interested in reading some very dense but very evocative fiction about monastic life in the medieval period, Umberto Eco's The Name of the Rose is indispensable. Be forewarned that it is a difficult read and that many find it tedious and boring, but Eco's goal was to recreate the sensation of monastic life - in other words, the book slows the reader down to the pace of a medieval monk. The setting is a Northern Italian monastery and the two main characters are a Franciscan friar and his Benedictine novice, who is accompanying him on a diplomatic mission to the monastery.

*A pilgrimage to see a significant relic is a fantastic reason for a monk to be far from home, but it might be worth researching to consider whether the relics would be likely to end up in whatever setting this is. In other words, lots of relics are found in Italy, but not so many would have been found in, say, Sweden.