Composers: From 1940 to 1958, Scott Bradley was the sole composer, not only for Tom and Jerry, but for MGM animation in general (the only exception being 1952's "The Missing Mouse", which was by Edward Plumb). Steven Konichek was the composer for the Gene Deitch shorts. Eugene Poddany, Carl Brandt, and Dean Elliott worked on the Chuck Jones shorts.
Directors: All shorts from 1940 to 1958 were directed by William Hanna and Joseph Barbera. The shorts from 1961 to 1962 were directed by Gene Deitch (and are considered to be, at best, the strangest era of Tom and Jerry, and at worst, the...worst era of T&J). The shorts from 1963 to 1967 were by Chuck Jones, Maurice Noble (co-direction), Abe Levitow, Jim Pabian, Ben Washam and Tom Ray, most of whom were alumni from Warner Bros. (specifically, Looney Tunes).
Keep Circulating the Tapes: Mouse Cleaning and Casanova Cat have seen release on VHS, but not on DVD and likely won't be rereleased on blu ray either.