Approximately how thick is a single strand of cat (domestic housecat) hair?
There are a few different types of hair on a cat.  But I think you are referring to the down hair on the cat.  It will also vary on the breed of domestic cat.  As a general statement, you can find a lot of quirky characteristics of mammals by looking at early genetics papers where people where trying to find genetic elements to obvious phenotypes.

For the down hairs, you can see "The ‘rex’ type of coat in the domestic cat."
Journal of Genetics [0022-1333] Searle, A G and Jude, A C yr:1956 vol:54 iss:3 pg:506

Table 3. Thickness in µ of normal and rex down-hairs

[Type of Cat][Diameter in um][SE of Mean]

Normal          25.8;   1·32;

English rex     16·2;   0·54;

German rex      17·0;   0·50;