the obvious paoblem is that the concrete will be unstable and take much longer to firm because of the excess moisture. As far as structural problems there shouldn't be any. Just will take forever to set up.
It depends on the type of concrete. Concrete is a mixture of limestone and other componds. When you pour concrete you need to allow it to cure. If you don't allow your concrete to dry it can crack and is very weak.
If you add water via rain to the concrete mixture, it will dilute it to a consistency that risks not being thick enough to "set" firm. The ratio of powder concrete and water is not variable.
