Covering the Face for a Woman in a State of Ihraam:

Dear husband, your wife is a well-protected pearl that should not be seen by strange non-Mahram men. You are obliged to order her to wear Hijaab. However, in the state of Ihraam, she must not cover her face. The Prophet said: "A woman in the state of Ihraam must not cover her face or wear gloves." [Al-Bukhari]

Since Ihraam for a woman includes her face, it is prohibited for her to cover it. Nevertheless, if there are non-Mahram men nearby and she needs to cover her face, she should lower a cloth to veil her face from above her head. ‘Aa’ishah narrated, "Men would pass us when we accompanied the Messenger of Allah while we were in the state of Ihraam. When they came by us, we would let down the outer garment from the head over her face, and when they had passed on, we would uncover our faces." [Abu Daawood]

Married Women Using Henna:

According to Sharh Muntaha Al-Iraadaat, it is a Sunnah for married women and not unmarried women, to use Henna upon Ihraam due to the Hadeeth narrated by Ibn ‘Umar which states that it is a Sunnah that a woman rubs her hands in Henna because it is considered adornment so it is recommended for her like perfume. Using Henna is disliked after this and as long as she is in a state of Ihraam as it is much like adornment.

Leaving Muzdalifah before Dawn for Those Who are Accompanied by Women:

From among the great virtues of the true religion is that it takes into consideration all different circumstances and states. This includes caring for the weak among the Muslims in Hajj by permitting them to leave Muzdalifah after moonset. Scholars considered women and children among the weak.

Shaykh al-Islam Ibn Taymiyyah said, "Weak people like women, children and the like may hasten to leave Muzdalifah for Mina after moonset."

Authorizing Others to Cast the Pebbles:

Casting the pebbles is one of the duties of Hajj. A pilgrim may not abandon it or authorize another person to do so unless he or she has an excuse like an illness or pregnancy. Crowdedness is not an impediment or an excuse. Therefore, either of the spouses may authorize the other to throw the pebbles on their behalf in case of an illness due to which they cannot throw.

Performing Hajj with a Child:

O spouses, if you perform Hajj with your children who have not yet reached puberty, you will be rewarded for this. However, this Hajj would not suffice the child of the obligatory Hajj. Ibn ‘Abbaas narrated that a woman lifted up a boy to the Prophet and said, "Would this child be credited for performing Hajj?" He said: "Yes, and you will have a reward." [Muslim] Ibn ‘Abd al-Barr said, "The scholars have agreed that if a child performs Hajj before puberty, then he/she is obliged to perform Hajj again if he/she finds the means and has reached puberty. The Hajj he/she performs after puberty avails him/her of the obligatory Hajj, and the same applies to ‘Umrah."

Spouses Performing Hajj on Behalf of each Other:

It is permissible for a man to perform Hajj on behalf of a woman and vice versa.

O spouses, death may strike either one of you before fulfilling the obligation and performing Hajj to the Sacred House of Allah. In this case, is it obligatory upon one of the spouses to perform Hajj on behalf of the other?

It is not obligatory, however, it is recommended that a husband performs Hajj on behalf of his wife or authorizes someone to perform Hajj on her behalf and vice versa because this entails reward and good treatment.

Shaykh Ibn ‘Uthaymeen was asked, "Would the husband be rewarded if he authorizes someone to perform Hajj on behalf of his wife who died without performing Hajj?"

He answered, "It is better that he performs Hajj on her behalf himself in order to perform the rituals in the best possible manner which he loves…but this is not obligatory." [Al-Liqaa’ Ash-Shahri (34) question number (579)].

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References:

Al-Mughni by Ibn Qudaamah

Al-Kaafi by Ibn Qudaamah

Haashiyat Ar-Rawdh by Ibn Qaasim

Kash'shaaf Al-Qinaa‘ by Al-Buhooti

Sharh Muntaha Al-Iraadaat by Al-Buhooti

Rawdhat At-Taalibeen by An-Nawawi

Haashiyat Ibn ‘Aabdeen

Al-Kaafi by Ibn ‘Abd Al-Barr

Fataawa of the Permanent Committee in Saudi Arabia, Ibn Baaz and Ibn ‘Uthaymeen