Do you know what kind of material they used to draw or paint in this sketch?
The Ancient Egyptians typically used mineral-based pigments like carbon black and ochre red. Iron oxides were also used for reds. As far as tools: reed brushes were common.
The sketches give you a window into daily life. The cat and mouse were probably just for fun, but this image of the god Osiris may have been an artist's sketch while preparing for a larger piece or just practicing.
Would this "sketch" have been made to be left as is, or would it be to guide the sculptor in carving out the figure?
It doesn't seem to have been a preparatory sketch for a sculpture, as it is missing the grid that artists used to block out the figure.
I thought that grids were just used to copy an image from one surface to another? Did they draw images onto stone before carving it?
Ancient Egyptian art was so regulated with such an emphasis on perfection that artisans used a grid system to lay out the proportions of figures in formal compositions. As the term sketch suggests, this is likely an example of an artisan's practice drawing.
Sculpted images were certainly drawn first. There is a great example of an unfinished relief in the Mummy Chamber.
Sometimes, artisans did simply paint 2-D images onto stone.
Thank you. I didn't know that.
Would the painting without carving have been a cost saving method or an artistic choice?
It would have been a choice made by the person who commissioned the work. The more intensive the work process, such as relief carving, the more expensive the work would be for the patron. Funerary equipment, for example, could be commissioned specifically or come though a more "off the rack" fashion. Much of what we seen in the gallery today would have been created for more wealthy patrons.