Q. 1. I heard that in the early days of the Church, the Sacrament of Baptism used to be administered in the nude. Is that true?
A. 1. Yes, it is true. In the early days of the Catholic Church, catechumens were baptized in the nude.
Quoting from Wikipedia, The Free Encyclopedia, it states:
" The early Christian Church reflected contemporary attitudes towards nudity, where it was considered acceptable in some contexts such as working outdoors. For example in John 21:7 Simon Peter is naked while fishing from a boat, but then gets dressed before meeting Christ. Additionally, in the Old Testament both Isaiah in Isaiah 20:2-3 and King Saul in 1 Samuel 19:23-24 are described as preaching in the nude. But overall the Old Testament is not positive towards nudity. 
The first recorded liturgy of baptism, written down by Saint Hippolytus of Rome in his Apostolic Tradition, required men, women and children to remove all clothing, including all foreign objects such as jewellery and hair fastenings. Later Christian attitudes to nudity became more restrictive, and baptisms were segregated by sex and then later were usually performed with clothed participants. Some of the Eastern Orthodox churches today maintain the early church's liturgical use of baptismal nudity, particularly for infants but also for adults.
Early Christian art included depictions of nudity in baptism. When artistic endeavours revived following the Renaissance, the Catholic Church was a major sponsor of art bearing a religious theme, many of which included subjects in various states of dress and including full nudity. Painters sponsored by the Church included Raphael, Caravaggio and Michelangelo, but there were many others. Many of these paintings and statues were and continue to be displayed in churches, some of which were painted as murals, the most famous of which are at the Sistine Chapel painted by Michelangelo.
In 1981, Pope John Paul II expressed the Catholic Church's attitude to the exposure of the human body in Love and Responsibility: "The human body can remain nude and uncovered and preserve intact its splendour and its beauty... Nakedness as such is not to be equated with physical shamelessness... Immodesty is present only when nakedness plays a negative role with regard to the value of the person... The human body is not in itself shameful... Shamelessness (just like shame and modesty) is a function of the interior of a person."
 Knights 1999.
 Hippolytus 2013, p. 33.
 Wojtyla 2013."