Mutual Inductance – Self Induced Voltage

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.

Since the secondary loop is open I2 must be 0

Of course it is - there is no current only an induced voltage. Because there is no current, the capacitor on the secondary side is effectively short circuit. And, because there is no current in the secondary there is no part-current flowing in the primary that is due to secondary current.

This boils down to a simple transformer problem where you need to find the voltage on L (primary) then knowing the coupling factor (aka M) you can calculate the secondary voltage.

Can you take it from here?