It should be fairly anti-symmetric, since the dynamics of losing are similar to those of winning. So instead of exponential looking, it's shaped like exp(rank) - exp(-rank). (just confirmed this with a quick simulation)

With participants able to go into debt indefinitely, the process becomes a random walk.

The result should be each participant being up or down from 0 at a rate of proportionated sqrt of t, the number of iterations of the system. Implying a wealth distribution which would become ever-more skewed over time.

Removing the boundary doesn't really make sense -- if you allow people to go into negative dollars but still be able to give a dollar each tick, then there's no longer a finite amount of money in the system.