I just went through pretty much the same thing this past weekend (here in London, Ontario, Canada). We had a BIG thaw over a couple of days and a LOT of snow on the ground. In my case, the sump pump worked fine, but the outside discharge pipe froze up overnight, so the pump could not move water the next day when the temperature rose again. While the sump well did not overflow, the hydrolic pressure around the house foundation was really high, which forced puddles to form in my basement wherever there was a crack. It was a pain in the ass, but I was pretty amazed that the pressure was so high.
Anyways, to answer your questions: That your sump well is filling every 5 minutes is not super-unusual - unless it continues for hours. There's a LOT of water in the ground flowing into your weeping tiles. One thing you'll want to check is that your outside discharge is sending the water at least 10 feet away from your foundation (I send mine 25 feet away) and that your lot grade is flowing the water away from your house. When I moved into my current house, the sump discharged literally 2 feet from the house foundation, so a heavy rain/melt would ensure the sump ran constantly.
A few other of my lessons learned over the years: Keep a spare sump discharge (flexible) hose handy. If your discharge freezes or breaks, you'll only notice it at the very worst time. Having a spare is a life-saver. Second, keep a spare float switch handy for your sump pump. Again, if your existing switch stops working, you'll only notice it at the very worst time. A spare only costs a few bucks and is another life-saver. Last one - consider disconnecting your your sump discharge hose in the winter (unless you can ensure it has enough slope to avoid freeze-up). That avoids the freeze-up issue (thank god we were home when ours froze), and if you get a big thaw in the middle of winter you can always reconnect it. But if you get the big thaw and no one is home, your sump pump can at least move the water from your basement (if if it means the pump runs constantly for awhile).
