The equation is parabolic since the determinant of the characteristic matrix is 0 . How can I justify this ?
I don't know any proper method to find the number of IC's and BC's required for a problem. For me it's equal to the order of the highest derivative. Therefore, total 2 BC's or IC's together.
First, in order to determine the classification of a second order linear PDE in two variables, you let $A$ be the coefficient of $u_{xx}$, $C$ be the coefficient of $u_{yy}$, and $B$ be the coefficient of $u_{xy}$. Then, calculate $\Delta = B^2-4AC$. When $\Delta$ is positive, zero, or negative, the equation is hyperbolic, parabolic, or elliptic, respectively. In your case, $B$ and $C$ are both $0$, so $\Delta = 0$. Thus, the equation is parabolic.
Second, the number of conditions needed for each independent variable is the number of conditions needed for the ODE created by considering all other independent variables as constant. Taking $y$ as constant gives a second order equation in $x$, so you will need 2 conditions where $x$ is known. Taking $x$ as constant gives a first order equation in $y$, so you will need 1 condition where $y$ is known. Thus, the total number of conditions will be 3. Whether you classify these as boundary or initial conditions depends on whether each independent variable is classified as a space or time variable, respectively. Usually, $x$ and $y$ are space variables, so in this case, we would have 3 total boundary conditions.