Question: Does a Hanafi who travels three days' way [104 km] and plans to stay for a week at his/her destination become a safari (traveler) automatically?
Yes, deciding to go somewhere is considered niyyah (intention).
Question: It is written in the book Se'adet-i Ebediyye, "Unless one makes intention, one does not become a safari even if one travels all over the world." It is stated that there is no niyyah without safar (travel), and there is no safar without niyyah. I understand from these statements that if I fly to the USA without making intention I will not be a safari. Is what I understand correct?
It is incorrect. You buy a plane ticket to go there, get on the plane, and fly there. Are these acts not considered niyyah (intention)? Do you not do them of your own free will? Niyyah means deciding to do an act. You say that you are going to the USA but at the same time say that you have not decided to go there. Actually, your going there indicates your decision. You have both intended and set out for the journey.
After the statement you mentioned, it is written in Se'adet-i Ebediyye, "An example of this is the case of soldiers in quest of the enemy. But they will become safari on their way back." The explanation for these statements is as follows: In this example, soldiers try to find the enemy around the place they are in; they do not try to find them at a place that is at a safar distance. However, if they reach a place that is at a safar distance, they will not become a safari because they did not intend to go there.
Question: If one goes to the city of A that is at a safar distance with the intention of staying there for 30 days and then if one, after staying there for 10 days, goes to the city of B that is at a distance of 10 km to stay there for 10 days and then goes to the city of D that is at a distance of 10 km to stay there for 10 days and then returns to the city of A, will one be considered a safari throughout one's stay for 30 days?
No, one will not be a safari (traveler), but a muqim (resident). When one becomes a muqim, one remains a muqim unless one travels safar distance from there. If one initially knows that one will stay for 10 days in the city of B and for 10 days in the city of D, one will be a safari during 30 days. In this case, a Hanafi who also follows the Maliki Madhhab in salat out of necessity would be a muqim throughout one's stay.
Question: As the city of Izmit is at a safar distance from Fatih, Istanbul, a woman cannot travel there without a mahram. An imam has made up a trick for this, which is as follows: "Do not make intention until Pendik [which is at a distance of 56 km] or Gebze [which is at a distance of 71 km]. Even if you make intention after reaching it, you will not be a safari, so you can go to Izmit without a mahram thereafter." Is it permissible for a woman to travel from Fatih to Izmit without a mahram by applying this method?
It is in no way permissible. Then why would she make intention in Gebze? She would travel until Izmit without intention. She would go to any city without intention then, and she would not be considered a safari even if she traveled all over the world without intention. In this case, there would be no rules regarding safar. The truth of the matter is that niyyah (intention) means deciding to go somewhere in one's heart. It is invalid not to make intention in order not to become a safari because it is niyyah to decide to go to a city and to buy a ticket for it. If one decides to go somewhere and buys a ticket for it but does not go there, one will not be a safari. That is, in order for a person to become a safari, both niyyah and the act of going are required. If one has decided to go somewhere and has gone there, it is invalid to say, "I did not make intention." This is not called hila ash-shar'iyya (legal stratagem aimed at finding a solution for a prohibited act), but hila al-batila (false stratagem).