Is a woman`s prayer considered invalid if non-Mahrams (marriageable men) saw her offering it?
A woman`s prayer isn`t invalidated if non-Mahrams saw her offering it, but she had better pray in isolation.
Which hour coincides with the last third of the night?
This hour could be figured out by calculating the number of night hours which start at sunset, and end at dawn break, then the total is divided on three to get the result.
What is the ruling on looking at pictures of naked women via the internet?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
It is impermissible to look at pictures of naked women via the internet, and this act is forbidden according to Sharia. The proof of this is that Almighty Allah Says (What means): "Say to the believing men that they should lower their gaze and guard their modesty: that will make for greater purity for them: And God is well acquainted with all that they do." [An-Nur/30]. Therefore, we advise you to busy yourself with offering acts of obedience that draw you closer to Allah and prepare yourself for meeting Him on The Day of Judgment. And Allah The Almighty Knows Best.
Is wearing a cap permissible for boys, or girls?
All Perfect Praise is due to Allah, The Lord of The Worlds.
In principle, anything is permissible sol long as there is no Sharia evidence prohibiting it, thus wearing the cap by boys is permissible as long as it isn`t done in imitation of the disbelievers. Moreover, a Muslim woman should avoid all that which attracts the attention of men because whatever leads to the forbidden is forbidden. And Allah Knows Best.