I guess the well has borders at 0 and a. Hence we know (with 100% probability) that particle is inside the well. Thus probability that particle is inside the well is 1:
There must be some more information.
There may be several ways to determinate with respect to the initial conditions.
I disagree. I've showed the way to calculate c, frankly speaking what we have here is an initial condition. We can calculate expressing as linear combination of eigenfunctions.