Yes, either proving a true conjecture or refuting a false one is "solving" it.

The Mizohata-Takeuchi conjecture is a statement in the form "For all <x> (a bunch of math)".

Showing that there exists an x such that the statement is false disproves the conjecture.

She found a counterexample.

She found more than one way of disproving it in the process.

Yes, found a counterexample to the conjecture.