Can every transcendental number be expressed as an infinite continued fraction? Every infinite continued fraction is irrational.  But can every number, in particular those that are not the root of a polynomial with rational coefficients, be expressed as a continued fraction?
The infinite continued fractions are precisely the irrational numbers; you will find a proof here, along with a proof that the expansion is unique.
The set of non-computable numbers is a strict subset of the transcendental numbers. If infinite continued fractions are computable, then I believe that they would not include all the transcendental numbers. Chaitin's constant is transcendental, but cannot be represented by an infinite continued fraction. Chaitin's constant cannot be represented through an algorithmic process.