My question is about the length of the inch which is a subunit of the Imperial foot. Is there any connection whatsoever between the Imperial system for units of measure and the dimensions of the earth? If so, what is the relationship?
It goes back to the ancient Romans.
The Romans, when working with fractions, often divided things into twelfths rather than tenths (since twelve divides nicely into thirds as well as quarters, whereas ten only nicely divides into tenths and fifths which aren't as useful as thirds and quarters). So the Roman foot was divided into twelve smaller units called "unciae" (singular: "uncia"), whence the English word "inch" (though the Roman uncia was slightly smaller than the Imperial inch).
In regard to measuring the earth, there are five Roman feet in a pace, and a thousand paces in a Roman mile (the English word "mile", in fact, comes from the Latin word "mille" which means "thousand"). So, one Roman mile was 1000 paces, i.e. 5000 feet (again, this was slightly smaller than the Imperial unit).
None of these units are directly relative to the Earth's size, though. They're only relative to other units.
At first an inch was the width of a man's thumb. In the 14th century, King Edward II of England ruled that 1 inch equalled 3 grains of barley placed end to end lengthwise. So no, no relationship to the dimensions of the Earth.