RebYidd, there is a machloket between Rambam on the one hand and Rabbenu Tam and the Meiri on the other as to whether shituf is permitted to gentiles. If it is it might not be a problem for them to daven to Yoshke. For us though it is avoda zara. That includes davening to a tzaddik (Gesher HaChaim 2:25 – http://www.hebrewbooks.org/pdfpager.aspx?req=31172&st=&pgnum=196(?
Joseph, regarding going into a mosque, google “going into a mosque” and you will find different opinions.
Zahavasdad, that is a PA lie. The Maharat HaMachpela is a Jewish holy site with a separate area for Moslem prayers.