I would like to know the jobs of a Scullery Maid in the early 1900s and if they had jobs outside the kitchen? Were they even allowed to leave the kitchen?
Scullery maids were often young girls and the lowest ranked among female servants.
In large households, scullery maids were hired to assist the cook. Scullery maids, who were usually very young girls, were also the lowest-ranked of the female servants. As such they were the first to wake up in the morning and the last to go to bed, worked very hard and were looked down upon even by the other servants. They didn't even eat at the communal servants' dining hall table, because they had to stay in the kitchen to keep an eye on the food that was still cooking.
It was a hard life.
Undoubtedly it was the scullery maid who was the most long-suffering individual below stairs. Penned up in a smelly scullery; bitterly cold in winter stifling in summer, she worked her way through mountains of washing-up. Lead-lined sinks were used for washing crystal and delicate china. A mixture of lye soap and fine sand to scour saucepans was cruelly hard on her hands. Scullery maids had to get up in the middle of the night to refuel the enormous kitchen range.
Their primary task was to assist the cook but they had other tasks as well as this excerpt from The complete servant by Samuel and Sarah Adams explains (cited here):
light the fires in the kitchen range, and under the copper or boilers, and stew-holes—to wash up all the plates and dishes—scour and clean all the sauce-pans, stew-pans, kettles, pots, and all other kitchen utensils; and to take care that all the latter are always kept clean, dry, and fit for use. She is to assist the kitchen-maid in picking, trimming, washing and boiling the vegetables, cleaning the kitchen and offices, the servants'-hall, housekeeper's room, and steward's room; and to clean the steps of the front door and the area. She makes the beds for the stable men—and generally fetches, carries, and clears away for the cook and kitchenmaid, and otherwise assists in all the laborious parts of the kitchen business. [Wages from 8 to 12 guineas a year.]
Her duties consisted of whatever the other staff (mainly the kitchen maids) thought fit within that department.
I highly recommend you watch the PBS series "Downton Abbey", to get a good idea about scullery maids in the early 20th century time period. It's a good portrayal of a period drama, and there are few characters that play scullery maid in the series. Daisy, the scullery maid, has a big role in the series. So you'll get an insight into her duties as a scullery maid.