These are the kind of things you want to ask the previous owners about, BEFORE you move in. They know all of that information. When we left our last house, I gave the new owners a complete owner's manual to the house, with all of the things they would need to know to maintain it.
My first guess is this is a drain INTO the sump, possibly coming from a french drain somewhere near the foundation that had no place to go otherwise. To be more sure, trace it back to its source.
The second possibility lies in the fact that your sump lid is covered over, and appears to be sealed. That pipe may well be part of a radon mitigation system, sucking air (with radon in it) that comes into the sump from under the floor slab.
To know which of these it is, you need to check to see where that line goes to. If it exhausts somewhere, with a powered fan in the line, then it is radon mitigation. If it goes into a cellar wall, it is probably a part of a french drain.
That the pump was running frequently simply means there is a lot of water coming in.
The power line seems fine. Best is if it goes to a GFI breaker, but many older homes do not have these things.
If this IS part of a radon mitigation system, then you need to be careful when you open up the sump. These things work by creating a region of negative pressure. If you damage the seals, then it will work poorly, leaving you are more risk if there is indeed radon coming into your basement.
Find out what you have, what these lines do, BEFORE you mess around down there.