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).
2.singing
3.picking
4.realizing
5.raining
start всегда с окончанием ing
begin можно и так и так