low temperature allows superconductivity. Heat increase would move the conductor out of the superconductivity temperature range.
Heat is produced in a conductor via current flow and resistance. I(squared) x R.
If R=zero, then no amount of current could produce heating which would move the conductor into the non superconducting temperature range.
The article states that magnetic fields are the culprit.
Since large current flows produce large magnetic fields, then, yes. However it would not be the current which does it, but the resulting magnetic field.
There are four basic parameters that influence a superconductor: temperature, magnetic field, current density, and mechanical stress. There is a critical value for each of these above which the superconductor cannot be in the superconducting state. That critical value is a function of the other three variables. So increasing the temperature lowers the critical values of the other three. Similarly, increasing the current decreases the critical values of the other three.