What is the method of prayer for the traveler—in terms of shortening (qasr) and combining (jam')—in the mentioned situation, and is shortening and combining prayers during travel obligatory or recommended?

If Baghdad is your homeland, the ruling of permissible shortening of prayers (qasr) ceases upon your arrival there, and you are obligated to perform the full prayer (tamam). However, when you travel to Oman, you are allowed to shorten and combine prayers (qasr and jam') provided that the journey is permissible. The shortening of prayers begins after you have left all the houses of Baghdad. If you intend to stay in Oman for two weeks, it is not legislated for you to combine or shorten prayers during this period. However, it is permissible to do so on your way back. Shortening prayers is a Sunnah, and it is best to perform it. As for combining prayers, it is permissible, but it is better not to do so.