The problem you describe indicates that most if not all of the health problems you indicate are associated with your house. The variety of symptoms you report is unusual. The respiratory symptoms would be consistent with exposure to allergens such as mold, dog, cat, dust mite or some type of irritant chemical. The eczema appears to be due to a contact dermatitis and is likely to be independent of the other symptoms. The carpet is a prominent suspect.

The only contaminant that I know of that has been associated with sleeping problems is formaldehyde (released from particleboard, plywood, medium-density fiberboard, cabinets and a number of biocides). Formaldehyde can also cause respiratory symptoms.

The hair padding underneath the carpet is not common. I assume that it is horsehair. If that's the case, I would expect that it has been treated with a biocide. Most biocides are not a problem, but some do release formaldehyde.

The paleness, gray coloring in your complexions is unusual (I have never heard of it before). It suggests that you may be exposed to something different from the normal type of problems one runs into.

Two aspects of your house are potentially significant, its old age and recent renovation, both of these have the potential for contributing to health complaints. Any house with that long history may experience water damage and mold infestation and other occupant-associated problems. Any time a building is renovated, a variety of potential contaminants may also be introduced.

You have asked if there are any blood or other tests that can be conducted to determine the cause of your problems. Such tests could include tests for common allergens such as mold, carboxyhemoglobin if one suspects carbon monoxide and a number of metals if one suspects exposure to metallic contaminants such as mercury, lead, arsenic, etc. Such tests should only be done if there is a good reason to do so and should be conducted in consultation with your physician. There is no test that can confirm exposure to formaldehyde.

The pronounced paleness, gray facial coloring as indicated previously is unusual. As such, it is likely to be a good clue as to the cause of illness. A trained occupational health physician (I recommend Dr. Robert McClellan who has practiced in your community and is nationally known) may be able to diagnose your health problems based on this and other symptoms. My initial thought is that biocides used with the hair padding or other materials present would be a good place to start.