^I know that Sunnis venerate him, I am not sure about the other sects. Some Sunni Muslims in Egypt actually visit the Coptic Orthodox Monastery -that is said to be at the site where the Lord and his mother visited- and pray there. Isa could probably tell us more about this.
Yes. Its not uncommon for Sunnis, Shias, Sufis, and several other sects to venerate Christian saints or visit Christian shrines (although its not really accepted by many Islamic theologians/shaikhs). Muslims do venerate Christian shrines such as St. George Church in Palestine, Mar Sharbel & Mart Rafqa Shrine in Lebanon, Our Lady of Lebanon Shrine, Church of the Theotokos of the Life-Giving Spring in Constantinople/Istanbul, St. Maryam Assyrian Church/Shrine of the Magi in Iran, and many many more chruches and shrines.
I've heard that in Russia, some Muslims from the Caucasus region visit Orthodox churches. Also, one time there was a Russian Chechen (or Dagestani?) Muslim person who would visit the church I used to go to because they said that it made them feel at home.
In India/Pakistan, some muslims visit Sikh holy shrines.