Electronic – Transformers confusion with output voltage and number of iron cores used

To answer your questions: We don't choose anything with transformers to have positive polarity. See the wiki. You could reverse the dots on both sides and still get the same result. The dots help you to know what the physical relation of the transformer coils are. If you have two coils, you have four terminals to connect to the circuit. But the circuit doesn't care which side is positive or negative. It only cares if the coils are reversed or not. It is just a notation. Notations are only useful if you know the rules behind them and how they are realized in the model and the physical world.

As far as magnetic flux goes, you really need to think about the magnetic fluxes in terms of mutual inductance. The transformer model uses 4 inductances, 2 of which are mutual (usuually called M) and characterize the flux between the two inductors. Vp always exists across the primary, you have to have a voltage to have current, it may be zero if its a time varying field.

Maintaining high flux in the core applies to non-ideal transformers. An ideal transformer has no resistance in the coil and no magnetic resistance in the material of the core. The idea there is if you introduce a material you have losses, you also have losses to outside world. In a universe with only a transformer, all the field lines would loop back into the other end, and you would have no losses. In the real world they connect and do work on other materials. In addition, air is not a great way to conduct magnetic fields, it has a low magnetic permeability. So we use a good material like iron to conduct the most of the magnetic field with its high magnetic permeability. This also has its problems because iron saturates, it has a point where it can't conduct very well. This saturation must be taken into account or it will clip the top of the sine wave.

I think a better way to look at a transformer is you are trading off voltage for current. A power engineer might be concerned about the inner workings of a transformer where you have losses (and they actually have magnetic circuits with multiple coils) but in the ideal case all you have to worry about are the mutual inductance, the inductance of the coils themselves and the ratio of the coils.

And the last paragraph doesn't make much sense to me either. So I can't answer your question there.

This answer explains that for an unloaded secondary, the natural phase relationship between primary voltage and secondary voltage is is zero degrees.

It therefore follows that if there is a secondary load current (due to a resistive load), the current in the primary due to that secondary resistive load must be 180 degrees out of phase with the secondary load current i.e. as current flows into the primary, current flows out from the secondary.

This of course is for an ideal transformer and a resistive load.

If you ignore the leakages and magnetization inductance of the transformer, and the load is reactive, then there will be a 90 degrees phase shift.

Bringing in leakage inductance and DC coil resistance will/can muddy the waters. Bringing in magnetization inductance muddies the water a bit more.

The low frequency transformer equivalent circuit is this: -

As you should be able to see, if you considered all the leakages, magnetization inductance and losses and then added a semi-reactive load, the phase angle is quite complex to calculate.

However I do-not know, in which-way Lenz's law acts in transformer; because the law states the induced current will try to hinder the cause.

Strictly speaking, it is voltage that is induced and any current that flows is subject to the that voltage, the load and the leakage inductance.

but when the secondary circuit of a step-up transformer turned On (closed) , so-far I've know , the current in primary-coil goes-up

In normal usage, for a voltage transformer it is non-ideal to consider the secondary being short circuited. However, it makes no difference to the phase angle providing you obey the rules inherent to the model.