Is it required for a discerning child to make the intention at night if they wish to fast?
If a discerning child wishes to fast during Ramadan, they must make the intention at night because intention is one of the pillars of fasting, even though fasting is not obligatory for them.
When is Istikharah (guidance prayer) Du`a offered, before Tasleem (saying asalamu alikum warahmatu allah wa barakatuh to end the prayer), or after it?
It is offered after Tasleem.
Does collecting saliva and swallowing it break the fast?
Swallowing collected saliva does not break the fast, but it is an unnecessary and meaningless act.
What is the ruling on using eye drops while fasting?
Eye drops do not break the fast, even if the taste reaches the throat, because the eye is not an open passage to the body cavity.