Is Lipschitz condition necessary for existence and uniqueness of a solution for a first order differential equation? Is the Lipschitz condition only a sufficient condition for existence and uniqueness of $\frac{dy}{dx}=f(x,y), y(0)=y_0$? Is it also necessary?

First, it's worth pointing out that for a Cauchy problem
$$
\begin{cases}
y'=f(x,y) \\
y(x_0)=y_0
\end{cases}
$$
we require  $f$ to be a Lipschitz function only in the second variable, while it is enough for it to be continuous in the first one (this result is known as Picard-Lindelöf theorem).
A typical example for which uniqueness is lost is given by $f(x,y)=\sqrt{y}$ (which is indeed not Lipschitz).
As already pointed out, Peano's existence theorem (together with other more general results) guarantees existence if $f$ is just continuous in both variables.
Long story short: for a general ODE of the form $y'=f(x,y)$ being Lipschitz in the second variable is necessary for uniqueness, but not for existence.
