Does an isosceles triangle always have three congruent angles?

Isosceles triangles usually have two congruent sides, but the rule is that they actually have at least two. That means that they can also have a third congruent side. That means they are both equilateral and isosceles*, which I personally think is way too confusing, but that's how it works.

Example: A triangle has angles of 60 degrees, 60 degrees, and 60 degrees. It is both isosceles and equilateral.

*I think that equilateral triangles are actually a type of isosceles triangle, so that if you're asked on a math test, for example, whether a triangle is scalene, isosceles, or equilateral, you'd say equilateral.

No, Isosceles is two equal sides, although an equilateral triangle CAN be an isosceles triangle. And Angles of an isosceles triangle are not known (given) - simply two equal sides.
Three, like every other triangle.