All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
1- As regards making up the Witr Prayer, if you are in the habit of praying three Rak‘ahs for Witr, then you could make it up with an even number of Rak‘ahs, i.e. four, as we have previously clarified in Fatwa 83483; and you may conclude each pair with Tasleem if you so like, or pray the four continuously with two Tashahhuds and one Tasleem in the end, as in the Thuhr Prayer if you so like, or pray the four continuously with one Tashahhud in the end if you so like; but this form is not preferable.
2- As for what your friend has told you, if he means that this recitation as such is obligatory, then this is wrong as nothing is obligatory to recite from the Quran in prayer except for Al-Faatihah. However, if he means that it is legislated, then this is right. In this case, you may recite Al-Faatihah, followed by Al-Ikhlaas, and then Al-Mu‘awwithatayn. You may also recite Al-Faatihah and Al-Ikhlaas alone if you so like.
3- Concerning the way to purify the garment from impurity, we have previously clarified it in Fataawa 90184 and 83760. The way you have mentioned is sufficient, for your washing it in the end with water removes the state of being impure.
4- Finally, it is permissible for you to perform Ghusl while standing under the shower instead of taking water from the bowl. For further information, see Fatwa 91182.
Allaah Knows best.