Has the concept of non-integer $(n+m)$-dimensional spacetime ever been investigated by theoretical physicists?

One example of such an approach is Ambjorn and Loll's Causal Dynamical Triangulations, which is very similar in many ways to the very old idea of Regge calculus, whereby spacetime is discretized. At small scales, non integer dimensions can emerge. For an introductory article , see

Jan Ambjørn, Jerzy Jurkiewicz and Renate Loll. The Self-Organizing Quantum Universe. Scientific American (July 2008), 299, pp. 42-49. doi:10.1038/scientificamerican0708-42, available here.

yes, in DIMENSIONAL REGULARIZATION dimension is just a parameter and after calculations you set it to $ d=4-\epsilon $ with epsilon tends to 0 so the poles of the Gamma function $ \Gamma (s) $ appear

curiosly enough, if we lived in a world with $ 4.1 $ dimension, then the Gamma function wuold have no poles and the Quantum gravity would be renormalizable.

another question is could the dimension for high energies be only a 'parameter' to be fixed by experiments or by renormalization of the theory ??