I did a report on Einstein a couple years ago, and most all my sources said in 1903, however they all disagreed on the exact date.
No, Albert did not have children with Elsa Einstein.
Albert Einstein was married to Elsa Lowenthal for 17 years
Mileva Maric and Elsa Einstein
Elsa Lowenthal was Albert Einstein's second wife. She was a cousin who nursed him back to health following a serious illness in 1917. Elsa taught Albert the importance of health.
Einstein wrote daily to his wife Elsa and adopted stepdaughters Margot and Ilse.
Albert Einstein had gotten married to Elsa, and Meliva Maric
He had no children with elsa but they raised Albert children from his previous marriage 
June 2, 1919
she was special because she loved him They were cousins. When Albert got sick Elsa helped him. Then later in life they married!
Mileva Maric. After divorcing her, Einstein married Elsa Lowenthal Einstein.
He got married to Elsa in Jersey in 1919.
Albert Einstein's second wife, Elsa died in 1936. He was married once before that, to Mileva, and was divorced before he married Elsa.