Does a traveler pray the traveler's prayer or the resident's prayer if he intends to stay for a month? And what is the duration for shortening and combining prayers in this case?

If you intend to stay, you must perform the prayer in full, and you are not allowed to shorten or combine prayers, as shortening and combining are specific to the traveler. Malik narrated from Salim that Ibn Umar used to say: "I pray the prayer of the traveler as long as I do not intend to stay, even if I am detained for twelve nights." Anas narrated: "When the Messenger of Allah, peace and blessings be upon him, set out before the sun declined, he would delay the Dhuhr prayer until the time of Asr, then dismount and combine them. If the sun declined before he set out, he would pray Dhuhr then mount."