I had trouble with the sump pump - it kept on running. I disconnected the water outlet pipe that ran from the pump to the pipe that runs through the wall so that I could run the water through the window to outside. Immediately water ran back in through the pipe in the wall for about 5 minutes, once that stopped all the pressure from immediately around the house had run in and been pumped out and everything was peaceful again. Obvuiously this is a tempory measure and it has to be fixed. The pipe through the wall runs underground (which is frozen in the winter) and I have no idea where it emerges. I am assuming that the problem is right by the house, because the water immediately ran back in. Can this be fixed or should I start again? I dug down and I notice that when the pipe comes out of the wall it is wrapped in a silver foil.

Sounds like a broken pipe from freezing. Biggest problem is another freeze. You need to go below the freeze line and make sure the wall joint doesn't freeze either. Not sure of the codes in your area but If you have enough slope or outfall, bring the pipe out of ground 30-40' away from the house and let it drain out. Make sure water drains away from the house. People pump their water into gardens, dry wells, watering trees etc.

Check local codes and see if it is allowed in your area to tie the sump pump into the house sewer. Many areas do not allow this, however, many do also. My area allows it, so this is usually the way we do it. If its allowed in your area, then it will be your best option.

The reason that some areas don't want sump water in the sanitary side is that they end up treating very clean water and it costs the taxpayers money. They might let you run it into the storm sewer but then you have the freezing issue.

In my area and climate, Illinois, we can run it to the back yard above ground or to the storm sewer for a hefty fee. Your pipe should exit the wall about 1' or more above grade. It should have a good pitch out toward where you want to direct it, it will not freeze even in our Chicago 10 below winters