Is a Muslim is Permitted to Pray Wearing Shoes?
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 for a Muslim to pray wearing shoes or slippers so long as they are pure, since a man asked Anas Ibn Malik: "Did the Prophet (PBUH) use to offer the prayers with his shoes on?" He said: "Yes." Moreover, Ibn Rajab (May Allah have mercy on his soul) said: "Performing prayer wearing slippers/shoes is permissible according the majority of the Muslim scholars." [Fatih Al-Bary,vol.2/pp.274]. And Allah Knows Best.
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 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.
Is it permissible for us to sever ties of kinship if our blood-relatives` gatherings/meetings involve acts of sin?
Observing kinship ties is obligatory, and paying your blood-relatives a visit causing their sinful acts to be hindered , then you should do so. However, if their sinful acts continue while you are at their gathering and they didn't respond positively, then observing kinship ties via telephone and the like will suffice in this case. And Allah Knows