If the transformer was designed for the specific frequency in use, it will step up or down voltage and current as it was designed to do. Transformers transform how power "looks" by increasing and decreasing voltage and current, while keeping power output equivalent to input (if you ignore the transformer losses).
If dc current is passed through a single coil, it is going to remain constant.
A current would be induced in the coil.
Basically, if you fill a room with steam and pass an electrical current through it... Does anything interesting happen?
Any electrons flowing through a superconductor will show up as a regular electric current.
pass a magnetic field through the coil of wire.
Yes, we can boil water if we pass enough current through it......
Yes.
Resistance in the circuit makes it difficult for current to pass through.
If the nail is iron and you pass an electric current through the wire then you have an electromagnet.
magnetic flux will be induced its use to rotate the shaft or given some force
it can easily pass through a wire a conductor or a battery
DC current will not pass through a transformer, in the sense that a DC current on the primary will not produce a DC output from the secondary. A constant DC current will not produce any output from the secondary but there may be transient effects as the DC current is connected or disconnected (in which case, it's really an AC current, isn't it?) More about this below because I don't think that's really the question. If you pass DC current through either winding of a transformer, two things will happen. First, you will heat up the transformer and, if you have enough DC current, you will burn it out. Second, you will induce a magnetic field in the core. The more current, the closer the core gets to "saturation" or the maximum field it can support. This is important if there is both DC and AC current present because the more DC field in the core, the less core capacity is available to "transform" AC current. As the core operates closer to saturation, the AC waveform will be distorted and some of the AC power will be lost to heating the transformer. For example, if the primary of a transformer is connected to an AC source, and a DC source is connected to the secondary, then the primary current drawn by the transformer will increase, possibly enough to destroy the transformer. There are special transformers, called magnetic amplifiers, which take advantage of this effect to use a DC current to modulate an AC current. There are transient effects of DC currents in a transformer winding. As the DC current magnetizes the core, energy is stored. When the DC current is disconnected, this stored energy wants to leave the core. It can do this by inducing a voltage in either of the windings. If both windings are open circuit, this voltage can be very high. So you may see a spark jump when the DC voltage is disconnected. In a large transformer, this discharge may break down the transformer insulation and damage it. Some switching power supplies take advantage of this effect in which case the transformer is wound slightly differently and called a "coupled inductor."