Is it permissible to perform Tayammum for a month due to a medical condition that requires avoiding water?
Tayammum is permissible so long as the above excuse lasts. And Allah Knows Best.
Is it permissible for the guardian, or the father of the child to hit the latter in order to teach him/her good manners?
It is permissible for the guardian to hit his/her child for teaching them good manners once they become ten years old provided that it is done mildly, harmlessly and without intimidation. Actually, such punishment is to be employed gradually i.e. after: asking, warning, and scolding. As for the teacher, he isn`t allowed to hit the student unless the latter`s father approves of it, and it is done according to necessity without intimidation, or harm.
Is Zakah (obligatory charity) due on articles of merchandise even if they were stacked in the merchant`s stores for years?
Yes, Zakah is due on goods even if they were stacked in merchant`s stores for years, in this manner Islam struggle against monopoly.
Who should pay the Zakah (obligatory charity) from the orphan`s money?
His/her guardian, and if the latter fails to do so, then the orphan himself/herself is obliged to pay the Zakah once he/she reaches puberty, and is capable of handling his/her own affairs. And Allah Knows Best.