Is photography forbidden, such as photographs of children?
All perfect praise be to Allah the Lord of the Worlds. May Allah`s peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
If there is a need for taking photographs of children or adults, then this is allowed. However, taking photographs of women shouldn`t be tolerated. Rather, they must be taken while wearing Hijab and must be kept in safe hands. And Allah The Almighty Knows Best.
Is it permissible for the young woman to turn the suitor down if she had no feelings for him?
Yes, it is permissible for her to do so.
Does offering Udhiyah (Offering a Sacrificial animal at Aid Al-Adha) avails from offering Aqiqa?
Offering an Udhiyah doesn't avail a person from offering Aqiqa because each of which is a different act of worship. And Allah Knows Best.
Is the Saum (Fasting) of someone who ate and drank forgetfully while offering fasting of oath expiation invalidated?
Whosoever eats, or drinks forgetfully is exempted by Allah; therefore, he/she shouldn`t break their fast whether it was obligatory, non-obligatory, or expiatory. And Allah Knows Best.