Checked out my 1956 two disc DVD set for references to the scenes you described. Moses' (Charlton Heston) beard does turn partially white after his first encounter with God at the close of disk 1. Later on in the film (disk 2 @ 1:15), an older Moses (now all white) receives the tablets, on which God does engrave the commandments on the tablets with fire. The tenth plague of the firstborn, is as I remember from past theatrical releases and televised showings. The parting of the Red Sea appears to have been green screened which might attribute some of the shots to looking like closeups, (of which some scenes actually are).
Nothing found for "two wheeler" races in The Ten Commandments, but just out of curiosity I checked my 1959 Ben-Hur DVD to see if there were any chariot races between Judah Ben-Hur and "Messala" as youths. Couldn't find the scene you described, possibly it exists in the 1925 release or later remake.