Is a fat-soluble vitamin a lipid?
Yes steroids are a class of lipids which include cholesterol and sex-hormones such as testosterone and estrogen. Only vitamin-D belongs to the class of steroids. 

Vitamins A, E and K are isoprenoids. In fact even steroids are synthesized from simple isoprenoids. You can refer to a standard biochemistry book to understand steroid biosynthesis pathway.

There are other classes of lipids too, such as fatty acids, phospholipids, ceramides etc. 

Yes all fat-soluble vitamins are lipids. This is the definition of Lipids from the IUPAC goldbook:


  A loosely defined term for substances of biological origin that are
  soluble in nonpolar solvents. They consist of saponifiable lipids,
  such as glycerides (fats and oils) and phospholipids, as well as
  nonsaponifiable lipids, principally steroids.


Both fats and fat-soluble vitamins are nutrients but are functionally different. Fats provide energy whereas vitamins perform regulatory activity and are required in little amounts in the diet compared to essential fatty acids.