Hi,
1. a) The optimal operating condition of a transformer in terms of efficiency is that its copper losses should be equal to core losses. The copper losses are the variable losses which depends on primary and secondary current while the core losses are constant for given frequency and dimensions of transformer
b) In order to achieve zero voltage regulation, the necessary condition is that the load power factor should be leading such that the p.f. = cos inverse (Xe2/Ze2)..............where xe2 and ze2 are the transformer reactance and impedance referred to secondary side.
Answer for 2, 3 and 4. The purpose of reactive power compensation in a system is to improve the power factor. If the power factor of a system is unity then, the system will be drawing only active power which is the power responsible for doing work, and if the power system is less than unity, it will be drawing reactive power, which is not utilized in doing real work but in changing the magnetic fields and results in heating of the devices and ultimately reduced efficiency.
Connecting a capacitor to an inductively loaded line provides the reactive power to the line which initially it was drawing from the generator or supply bus. Hence it draws less current from the supply and results in lower voltage drop which results in decreased voltage regulation.
6. Shunt inductors, Series capacitors , synchronous motors(with no shaft load and varying excitation) and static var compensators are some of the remedies used to reduce reactive power consumption of the line.