Before Gilmore Field, it looks like there was a small oval track there. This picture is dated 1935.
hollywoodphotographs.com
Regarding the soccer, I found that a friendly match was played at Gilmore Stadium in 1947. It was Los Angeles XI vs Hapoel Tel Aviv, and ended as a 1-1 draw. See
soccerstats.us for more details. The Wikipedia article I linked to in my first post above also says cricket was played there at least once.
