If a triangle has 2 sides of equal length, is it isosceles? I know that if the 2 angles of a triangle are the same, it is isosceles. But what if the two sides are the same? Can we conclude that the corresponding angles are the same and it is isosceles?
Yes. Assume $AC=BC$. Then triangles $ABC$ and $BAC$ are congruent (SAS), hence $\angle A=\angle B$.
Yes, you can use the fact that given a triangle A,B,C if $\alpha$ is the opposite angle of BC, $\beta$ the opposite angle of AC and $\gamma$ opposite to AB it holds $$\frac{BC}{\sin {\alpha}}=\frac{AC}{\sin {\beta}}=\frac{AB}{\sin {\gamma}}$$ but i bet there is a simpler proof that doesn't use trigonometry. By the way i thought that having two equal sides was the definition of isosceles; while it's equivalent i've never heard your definition.