Is there any research on approximation of reals with computable numbers
It's funny you should ask, because computability and diophantine approximation has actually been a popular topic in recent years. In particular Becher and Slaman and coauthors have many results and papers. See for instance

Becher, Verónica; Reimann, Jan; Slaman, Theodore A., Irrationality exponent, Hausdorff dimension and effectivization,  ZBL06837203.