1.) Yes.
2.) Should follow from a good pure AC sine wave excitation.
3.) Yes.

4.) The higher the permeability of the core the better - Mu metal is THE best.
5.) screen the primary and secondary with an interleaved grounded copper foil, or, failing that, make sure you have one end of the primary grounded, and that end is nearest the secondary.
6.) Multi-filar winding is absolutely vital, to reduce interwinding capacitance and ensure uniform flux over the windings.

They are capable of nulling out enormous reactive and resistive strays to the DUT. (>>1000pFof strays with <<0.01pF of signal.

Look up "Blumlein Bridge" on Google for some other implementation s. Named after Alan Dower Blumlein, inventor of stereo recording, and electronic television as well as the bridge that bears his name. Died in a plane crash, testing new British radar system "H2S" during the war.

Bit of a personal hero of mine.

Steve