Operation of Transformer Other Than Designed Frequency?

You were correct in most things. Shifting 50 to 60 gives 50/60 = 0.8333 on my calculator though! My textbook gives two components of the magnetising loss. Hysteresis loss is proportional to f x B x B^x (f= frequency, B = maximum flux density). The first f x B cancel each out because B falls in the same proportion as f increases. The B to power x has an x value between 0.5 and 1 for usual flux densities, hence B^x lies between 0.912 and 0.833. Clearly, hysteresis loss is reduced if a 50 Hz transfo is operated at its "50 Hz RATED VOLTAGE" at 60 Hz. The second component Eddy-current loss is proportional to B squared x f squared - here the fall in B exactly cancels the rise in f and the loss is unchanged between 50 and 60 Hz. In summary, magnetisation losses are less at the same primary voltage when a 50 Hz transfo is operated at 60 Hz. You may not be correct about volts per turn, you have not mentioned any change of primary voltage - so [since the transfo has a predetermined number of turns] the volts per turn will not change if the voltage stays the same. Regarding %Z, the thing to remember is that the leakage inductance is the same at both 50 and 60 Hz so its reactance will be 20% higher at 60 Hz. So roughly %Z will be 20% higher at 60 Hz. More exactly, remembering that the impedance Z is the long side of a vector triangle of which the other sides [at right angles] are reactance and resistance, the reactance has increased 20% but the resistance won't have increased as much as 20% so reactance will not have increased as much as 20%. The resistance will increase slightly at the higher frequency due to "skin effect" but this is probably negligible. The regulation might increase by 20%, but remember 20% occurs at a low lagging power factor while on resistive loads the voltage drop from no-load to rated load is small anyhow - so the effect for the load depends greatly on the load power factor. The only other factor is short circuit current - it will be 20% lower for 50 Hz transfo at rated voltage but 60Hz at 50 Hz will see 1.2 x 1.2 = 1.44 times design bursting forces in windings which is dangerous. Practically, whether the "fresh" transfo has the same rating as the original and whether the voltage ratio is the same will be just as important as frequency change. If the question was jacking an 11 kV 50 Hz transfo up to 13.8 kV, this would be "overcooking" since both flux density and frequency would be higher and no-load losses would increase a lot. Finally, one CAN run a 60Hz transfo on 50 Hz if you can drop the primary voltage 20%. For example, a 13.8 kV transfo could operate at 11 kV 50 Hz, but observing its CURRENT rating limit would drop kVA rating about 20%.