Different in degree. Dry skin just dry and flaky. It becomes "eczema" when it's itchy, red, oozy, crusted, blistering, swollen, etc.
Yes. People with eczema often complain that their skin is dry, but dry skin, in and of itself, is not the same as eczema. Treatments are completely different as well, so if you aren't sure what is going on, see a dermatologist.
Yes. Eczema is a form of hereditary dry skin that waxes and wanes and is more prominent in childhood. It is usually accompanied by cradle cap in infants and severe itching. Eczema has a particular pattern in the skin and it changes as you age, and mostly resolves in adolescence. People with eczema tend to have allergic problems.
Atopic dermatitis. Check out http://www. Mayoclinic. Org/diseases-conditions/eczema/basics/definition/con-20032073 for info on eczema aka atopic dermatitis which is a condition leading to red itchy skin which may or may not be dry. Check out http://www. Mayoclinic. Org/diseases-conditions/dry-skin/basics/causes/con-20030009 for info on (not so) common causes of dry skin like hot water, harsh soaps, sun exposure, etc.