What types of problems can be solved using slack, surplus, and artificial variables?
Well, that's a broad enough question. Where did you learn of these things? Perhaps the same source would have some examples?
It is very likely that these things were invented simply to solve a Simplex problem. It wasn't square enough and it needed more variables. Everyone does this when attempting to solve a problem and we begin to feel that our information is inadequate. We go looking for more stuff.
As is often the case, the things we invent of necessity eventually begin to attract our attention and we begin to see that they tell us more that we thought when we invented them.
Are you having a "Why do I need these things" moment?