1) Another moment and she began ( realizing; to realize) that one of the eyes of the portait started (to move/moving).
2) She's starting ( to sing, singing).
3) she overturned the work-box and began ( picking; to pick) up the articles one by one from the floor.
4) It took him twice to begin ( realizing, to realize) that such a detectives would never find his sister.
5) Take an umbrella. It's starting ( raining; to rain).

1.to realize,moving2.singing3.picking4.realizing5.rainingstart всегда с окончанием ingbegin можно и так и так