Is it permissible to shorten and combine prayers while staying at a friend's house before entering the hospital, and while staying in the hospital itself with the intention of staying for a specific period, and what are the implications for previous prayers if the action was incorrect?

You should have completed the prayers and ceased combining them as soon as you arrived at the city you traveled to, as long as you intended to stay for more than four days. This is the view of Malik, Shafi'i, and Ahmad. What you did, shortening and combining the prayers, was an error, and you must repeat the prayers. Abu Hanifa and those who agree with him hold that a traveler may shorten prayers as long as he intends to stay for less than fifteen days. According to this view, repetition is not obligatory if the duration does not exceed two weeks.