Church of the Nativity? Birth site of Jesus?

Do Muslims believe that Jesus is born in the Church of the Nativity (Grotto of the Nativity)?

Do Muslims believe of the Cave of the Patriarchs, as the burial sites of these couples? (1) Adam and Eve; (2) Abraham and Sarah; (3) Isaac and Rebekah; (4) Jacob and Leah

Do Muslims believe of the Wailing wall as a part of the Solomon temple?

Or other biblical sites.

If yes, do Muslims have their own proofs or citings in Islamic Books. OR hadeeth.

If no, do Muslims have their own opposing views, or their own beliefes of where the actual sites are?

Speaking only for myself I do not recall much reference to those in the Qur'an. Except that Maryam gave birth to Isa(as) beneath a Date Palm near a source of water. I still have much of the Sunnah to study, but so far have come across no mention.