Yes, you should dig to below the foundation at least 6". 12" is better. (I think I failed to mention this important item.) Anyway, the rock you found allows for water to displace air, and fill into the pipe. However, I'm just a bit alarmed that the slits are facing UPWARDS. They should be downwards. (See Constructing A French Drain.) if you're going to allow your cellar well drain into the existing system, that's fine but you ought to cover the pipe you exposed with filber fabric. The way I see it, water must rise ABOVE the level of the slits in order to enter the pipe. This method, unfortunately, will permit fine sediment to ENTER the piping, thus, clogging it up in due time. If you told me the pipe was wrapped in filter fabric, then I wouldn't be as concerned about clogging since the fabric will prevent dirt from clogging it. You may have to see what you can do to keep the piping from getting clogged if you don't have it fixed such that the perforations are facing downwards.
As for your last question, the perimeter system should drain away from the house, preferable to the street or a drain pit or somewhere where it won't 'collect' and eventually make its way into your home. If you're asking if I know WHERE it's draining, I can't say w/o seeing you home. (And not likely!) If it's a new home, ask the builder. If it's an existing home, ask the previous owners, or your current neighbors (if you're OK with them.) If you need more, post up.