Wave equation boundary condition I understand that:  
1.Setting a boundary condition of the type $$u(x_0,t)=0$$ means that the wave will reflect and change sign.  
2.Setting a boundary condition of the type $$\frac{du}{dx}(x_0,t)=0$$ means that the wave will reflect and not change sign. 
My question: What happens to the incoming wave if I set both types of boundary condition at the same point?
 A: General discussion
Any differential equation, either Ordinal or Partial, require as many conditions as the sum of highest derivatives in each variable. 
Examples:


*

*$\frac{d u}{dx}=f$ requires exactly one condition

*$\frac{d^2 u}{dx^2}=f$ requires exactly two conditions either 2 initial conditions, e.g. $u(x_0)=u_0, u'(x_0)=u'_0$; or two boundary condition e.g. $u(x_0)=u_0, u(x_n)=u_n$

*$u_t=u_xx$ (one dimensional diffusion equation) in finite domain $a\le x\le b$ requires one initial condition and two boundary condition one on each side.

*$u_{xx} + u_{yy}=f$ in rectangular domain $a\le x\le b$, $c\le y\le d$  requires 4 boundary conditions, one on each boundary. 


These conditions are used to determine the coefficients of the general solution to the differential equation. You solve a linear system of equations created using these conditions. If you got too many BCs you'll get overdetermined system (tall matrix instead of square) which may have no solution.
You can think about solving DEs as of integration and these constants are related to constants of integration. The BC's are sort of limits of integrals.
Regarding your question
One dimensional wave equation problem in finite domain may look like
$$
\begin{cases}
{u_{tt}} &= {c^2}{u_{xx}}&a\le x\le b, t>t_0\\
u(x,t_0) &= f(x)&a\le x\le b\\
{u_t}(x,t_0)& = g(x)&a\le x\le b\\
\left(\frac{d}{dx}\right)^n u(x_0,t) &= b_1(t)&t>t_0\\
\left(\frac{d}{dx}\right)^m u(x_n,t) &=b_n(t)&t>t_0
\end{cases}
$$
where $n,m\in{0,1}$.
You want to use $\left(\frac{d}{dt}\right) u(x_0,t)=0$ together with $u(x_0,t)=0$, but you can't put them together at the same boundary.
There is another sort of boundary condtion, a mixed one, i.e. Robin type. which looks like $a u(x_0,t)+ b\frac{\partial u(x_0,t)}{dx}$ a discussion about its physical interpretation can be seen here.
You may also be interesting about transparent boundary conditions, this is a wide family which is very useful for many things, for example for computation of wave scattering phenomena problem one uses a very special sort of this, an absorbing boundary conditions. You can read a little about this here.
