Let's only consider edges with positive probabilities.

Looking at a path with probabilities p1,p2,...,pn, its value is p1*p2*...*pn, we want to maximize this product. Taking the logarithm of the probabilities, we're left with maxmizing the sum log(p1)+log(p2)+...+log(pn) since the exponential function is strictly monotone. This sum is negative, if we multiply it by -1 we have a minimization problem. In short, take the negative logarithm of edge values, and then any shortest path algorithm (like Dijkstra's) will do the job for you.

Does negative probability exist at all? ;)

Gephi is dedicated to social network analysis and as far as I know, there are no plugins supporting this kind of computation.

We should find all paths from start to finish, for each calculate product and choose the minimum one. It is quite simple in terms of describing the algorithm but quite complex in terms of computation, especially for extensive graphs.

But as I mentioned Gephi here is useless, unless you write your own plugin. But if you have only one case to solve it would be easier to use e.g. python.

But the answer for your very first question is that you can assign values to edges in the Gephi, because the Gephi supports also weighed graph. Look at the "Data laboratory" tab, switch to edges and you will see the weight column that is able to store float values. This is exactly what you are looking for.