how to fix sticking wooden drawers?

Sticking drawers often result from worn runners inside the carcase. The wood grinds down over a century of use, letting the drawer drop and catch on the frame below. Rubbing a candle stub along the running edges helps reduce friction in mild cases. Wax acts as a dry lubricant without swelling the timber. Severe wear needs attention from a restorer. They splice fresh timber onto the worn surfaces to restore a level running action.

what are the keyholes made of?

Most mid-century keyhole surrounds are shaped from brass or turned from bone. Brass plates sit flush with the drawer front, pinned directly into the wood. Mother-of-pearl turning up on darker timbers adds a flash of contrast. Later decades saw stamped brass plates applied over the wood face rather than inset.

I bought a tall chest last week and it has this deep drawer at the top that seems far too big for folded shirts, what was it meant to hold?

That deep top drawer often held top hats or ladies' bonnets. Standard folded linen went into the shallower drawers below. Headwear required rigid storage to prevent crushing the felt or silk frames. Deep drawers provided a dust-free space away from the main bedroom traffic.

My aunt gave me one of these and the back boards are just rough pine planks with gaps between them, is it missing a proper finished back panel?

The back boards are supposed to look rough. Cabinetmakers rarely wasted expensive mahogany or polishing time on surfaces facing a wall. Wide pine planks shrank as central heating dried out modern homes, causing those gaps. The makers nailed them straight onto the carcase rebates while the wood still held moisture. A rough back indicates standard workshop practice for the period.