As stated in this answer, a regular-season game the night before Opening Day has been televised on ESPN since 1994(1). It is unclear if this makes Sunday Opening Day (as the first part of your question suggests) or Monday based on the statement "Major League Baseball still reserves the title 'Opening Day' for the first day in which multiple games are played(1)."
Specifically, in how many calendar years was the earliest game played during that calendar year across both the National League and the American League, on Easter Sunday?
In any case, since 1994, the earliest regular-season game has been played on Easter Sunday six times:
NOTE: Before 1994, dating back to 1914, the earliest regular-season game of each respective year was never played on a Sunday. It was usually played on a Monday, Tuesday, or Wednesday, with anomalies in 1915 and 1972, in which the earliest regular-season game of those years were played on a Saturday.