Question about ideal transformers and inductors
Lets's discuss only about what you said "I have understood"

Not sure how you consider those  those historical non-measurable quantities named EMF, but the result is ok for DC: There's linearly increasing current in an ideal coil which is connected to a constant DC voltage.

There's no need for the EMF. If some mechanism, say an ideal voltage source forces an ideal inductor to have voltage U between it's terminals, the current through the inductor has changing rate U/L where L=the inductance. This is how inductors work, otherwise it's something else than an inductor. No EMF needs to be imagined to exist.

With AC voltage source the same law can still be used. If the voltage has sinusoidal form, the current which changes with rate = U/L is also sinusoidal, but lags 90 degrees behind the voltage.

Some of us surely want to keep concept EMF alive because one can feel he's more systematic in that way. With EMF one can have a logical reason for the existence of current and measurable voltages between circuit nodes. In addition many clever looking laws can be formulated with EMF such as "in any closed circuit loop the sum of voltage drops must be equal with the sum of EMFs". 

But an alternative way is to discard EMF and  to formulate how some ideal components work with measurable voltages. Practical components can well be modelled as circuits which are made of ideal components. I applied that thinking when I wrote how inductors work.

For transformers we can write a pair of equations for primary and secondary currents and voltages. The equation pair states how it works and gives to us the foundation to find how currents and voltages in the whole circuit must vary in the time. There's no Federal nor other laws that force us to call the induced voltage components EMFs and tease us to include term EMF to our talks and writings before we can write the total law for a component. 

I heve learned to discard the EMF because everything which happens in circuits is finally a consequence of how electric and magnetic fields happen to settle as well inside the parts and around them. In Maxwell's equations there's no separate "motive electric field" and "drop electric field", there's electric field as a place and time dependent vector field.