Laudanum refers to a preparation containing opium such as tincture of opium. Gentian root refers to the rhizome and roots of Gentiana lutea , a yellow-flowered gentian of southern Europe used as a tonic and stimulant for the stomach. Spike oil has long been used in combination with turpentine for curing old sprains and stiff joints. Spike oil also kills lice and other external parasites on animals. Aqua ammonia is a water solution of ammonia.
Oil of turpentine is derived from the resin of pine trees.
Oil of turpentine is administered internally in cattle and horses to destroy parasitic worms and externally as a stimulant for rheumatic swellings. Oil of turpentine is also used to treat sprains and bruises, and as an antiseptic and vesicant. Sweet oil is usually either olive oil or almond oil. I don't know which is meant here. Eleanor Rockwell's farm hints courtesy Paul Lowrey.