Is Zakah due on the orphan`s money?

Yes, Zakah is due on the money of the orphan.

Is it permissible to pay the expiation of an oath to a charitable party as it feeds the poor and needy such as Tkiyet Um Ali?

All perfect praise be to Allah, The Lord of The Worlds, and May His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.

There is no harm on authorizing someone to handle the expiation of another to the poor and needy, since Shafiee's jurists stated that authorizing someone for the purpose of distributing Zakah, expiation and vow is permissible as stated in [Mughni Al-Muhtajj, vol.3 pp.237]. At last, the intention of the person who intends to authorize someone else suffices in this regard. And Allah Knows Best.

Is it permissible to make up for missed fast after the beginning of the second half of the month of Sha`ban(the month before Ramadhaan) ?

Yes, it is permissible, but one who had missed fasts should hasten to make up for them. As for the Hadith mentioned in this regard, the prohibition is for offering absolute voluntary fasting.

Am I permitted to perform more than one prayer within single ablution?

You are permitted to pray as many as you could within one ablution and to recite Quran so long as your ablution isn't nullified.