What is the ruling on Qonot Du`a (supplication) at dawn prayer?

All perfect praise be to Allah,The Lord of The Worlds                                                                                                                                                                        It is a confirmed Sunnah, and whoever leaves it should perform Sujood As-Sahw (prostration of forgetfulness). 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 Ramadan)?

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. And Allah Knows Best.

I am a disabled woman who has vaginal discharges and can`t control her urine. Therefore, I combine prayers. Is that permissible?

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 his family and companions.

According to the Hanbali doctrine, it is permissible for the sick to combine prayers. In your case (Asker), it is better that you delay the noon prayer until a few minutes before the afternoon prayer in order to minimize the time span between the two and perform each alone. In addition, you can do the same with sunset and evening prayers. Finally, we ask Allah to bless you with recovery. And Allah Knows Best.

Does an internal examination require ghusl?

Ghusl isn't required for making internal examination; however, it invalidates one's fasting. And Allah Knows Best.