Current input and output of transformer
1) Simple primary inductance does not limit transformer current. That gets short-circuited by the secondary.

What you need is leakage inductance. This is due to the part of the primary field that does not couple to the secondary. You can enhance leakage inductance by separating the primary and secondary coils, and placing soft iron shunts across the magnetic circuit between the primary a secondary.

A commercial variable current welding transformer makes these magnetic shunts adjustable, to vary the maximum current that can be drawn.

An alternative is to place an inductance, or resistance, in series with the transformer. As an experiment, or for use at a pinch, you could place an electric kettle in series with the transformer primary - water cooled, high power, mains rated, inexpensive, known current limit.

2) Neglecting primary magnetising current, which is usually small, primary and secondary current are linked by the transformer ratio.

3) You can liken primary and secondary currents to gears in a gearbox of suitable ratio, but they are directly linked. Flux is more complicated. It is directly linked to the weighted difference between the primary and secondary currents, AND its rate of change is directly linked to the voltage. I don't think the mechanical analogy is going to help you much here!