Are projective rings over $\mathbb{Z}$ free?

Even over a field this is a difficult problem! It is related, among other things, to the Jacobian conjecture and the constellation of problems around it.

In [D. Costa, Retracts of polynomial rings, J. Algebra 44 (1977), 492-502.] Costa deals with the two variable case (that is, with projective algebras generated by two elements) (he works with coefficient rings which are zero-dimensional and noetherian, for extra fun) He also deals with the non-commutative polynomial algebra.

Then [Picavet, Gabriel. Algebraically flat or projective algebras. J. Pure Appl. Algebra 174 (2002), no. 2, 163--185. MR1921819 (2003i:13009)] shows that over a field, a projective algebra of finite type is either a polynomial algebra or the coordinate algebra of a complete intersection.