For a traveler who intends to stay for seven days, what is the duration for combining and shortening prayers, especially with the presence of continuous secretions that necessitate Istinja' (cleansing) and Wudu' (ablution) before each prayer?

There is no necessary connection between combining prayers and travel. Among the excuses that permit combining prayers are: suffering from urinary incontinence, or a general need if one fears missing a flight or has an engagement that prevents performing the second prayer with it, or if doing so would be difficult. As for the duration of stay that permits concessions, the majority of scholars hold that one who stays for a week is not considered a traveler, and must complete the prayer and perform it at its due time, unless there is another excuse that permits combining, in which case he may combine while completing the prayer. Sheikh al-Islam Ibn Taymiyyah and Sheikh Ibn Uthaymeen differed from the majority, holding that a traveler remains a traveler as long as he does not intend an indefinite stay or settle in the country. The preferred view is that of the majority, and a layperson may follow whichever scholar he trusts.