Is the displacement d(x, t)=cx2+dt2, where c and d are constants, a possible traveling wave?

Ok, doesn't have the required math symbols I need to give you the best looking answer so we'll have to do this the best we can.  In short, yes it could be a wave.  A simple wave,ψ,  is described by the equation
d²ψ/dx² - (1/c²)d²ψ/dt² = 0.  The right side must be a nonzero constant, which is zero here.  This is a basic wave equation.  It says if the second partial derivative of a function with respect to space minus the second partial derivative of that function with respect to time, multiplied by a constant (1/c² here) equals a constant (0 here) then that function describes a wave.   doesn't have a partial derivative symbol so I used "d". Also the c in my equation is the speed of the wave.  Not necessarily the same c as in your equation.

Now let's look at your function.  The second partial derivative of it with respect to x makes the t term 0 (it's a constant as far as x is concerned) and the x term becomes 2c.  Doing the same thing for the partial derivative of it for t leaves 2d.  These terms represent two constants, which if subtracted, as the wave equation requires, would lead to a constant on the right, which could be made zero if the coefficients c and d are chosen correctly.  

It really doesn't matter for any of the questions that you've listed.

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