What are the risks and problems that starting a pullstart engine on a starter box will cause? (the pull start is still one the engine)

No risk and no problems. It will work just fine.

The only problem I can see is if the vehicle is not designed to use a box starter. For example, the Savage truck cannot be started with a box; it must be started with a pull start or roto start.

In addition, the Traxxas 4-tec (at least the older one with the TRX2.5) cannot be started with a starter box because the flywheel is too small. A larger flywheel would be needed in this case so the starter box wheel will hit it.

I guess it depends on what car you have as to if it is physically possible to use a starter box but as far as damaging your engine is concerned it is not a problem. I have a PS buggy that I always start with a starter box and have had no trouble at all.