Yes, there are molecules with axes of symmetry. Allenes are a prominent example. See this post. https://www.masterorganicchemistry.com/2015/01/13/chiral-allenes-and-chiral-axes/

Thank you, glad you are finding it helpful.

Yes, so long as the molecule doesn’t have double bonds that can be E or Z, which is another source of stereochemical difference.

e.g. (R)(E) and (R)(Z) would be diastereomers despite having only one chiral centre.