Welcome to the forum baw! I also leased a Leaf starting in August of 2012 and we received our 85kWh Model S November of 2012, so I will try to give you my experience having both cars. My reference points for my Model S charging are all from charging at the maximum daily trip level of 90%. There is still another 10% of charging capacity available for longer trips that I could charge up to in order to increase the Model S range. My Leaf reference point is charging at 100%.
1) Real world range - I live in the Seattle area, so we don't get extreme cold or much snow. My experience with colder weather was temperatures in the 30's to 40's as general lows in the winters. But we do get rain and windy conditions in the winter which can increase drag on the car from both the rolling and air resistance perspective. Your observations about the Leaf match my experience here in Seattle though. In the winter, I would only feel comfortable driving my Leaf 30 to 40 miles before getting a charge. I would not try to baby the car to maximize the range. I tend to just drive the car like any other car and be comfortable in the cabin with the heat on. With our 85kWh Model S, in my normal driving in the Seattle area, I never have any concern about the range I have. In the winter, with highway driving in the rain and cold, cabin heat on at a comfortable level our car our actual range is around 200 to 220 miles (We have the 21 inch performance tires on our car). So I consider Tesla's range estimates on their online calculator fairly accurate (we have a lot of hills in my area, especially where my residence is and I drive the hills daily, so that impacts our range quite a bit). The range impact of cabin heat and cold weather is not as high in a Tesla vs. a Leaf partially because the total range is so much larger in a Tesla. If you burn up 20 miles of battery range in the Leaf through cold outside weather and heating the cabin, that represents a significant percentage of the Leaf's actual 70 or so miles of actual range. If you burn up those same 20 miles of battery range in the 85kWh Model S and you have a actual starting range of 220 miles, you are not having nearly the same percentage impact on your usable range.
2) Heat comes out very quickly vs. Leaf. I would say with in 1 minute in my car if I don't preheat. But the iPhone app I have with the Model S is significantly better than the Leaf's iPhone app HVAC controls. The Tesla app allows you to turn the HVAC on and off as well as select the specific temperature you want. It also gives you feedback as to what the cabin temperature is (although my temperature readings are not accurate in terms of actual temperature because of issues with the temperature sensor placement in early VIN cars. I've learned to adjust the indicated temperatures to match an actual temperature in the car). With the Model S, you can be more comfortable invoking the preheat/cool functions compared to the Leaf because the Model S has so much more battery capacity. With the Leaf I would burn up 5 miles of range preheating/cooling the car if it wasn't plugged into the wall. That is a significant percentage of it's usable range (call it about 40 miles in the winter for me). With the Model S, if it burned up 5 miles, that is a small percentage of the 220 usable miles we have available. So the cabin heat works great in the Model S compared to my Leaf - warm air out of my Model S vents very quickly.
3) Yes, you can use your existing charger. The Model S comes with a J1772 adapter end. You stick that on your existing equipment to make it compatible with the Tesla charge port. No other modifications required. I would use a JESLA charger with a J1772 end for both my Leaf and Model S in my garage and it worked great. a 240V/30A charger will return about 17 to 20 miles of range per hour of charge on the Model S. Your Leaf sounds like it had that 6.6kW charging system, so you probably will experience a similar charge rate with the Model S as you had with your Leaf. For most commutes, that should be more than plenty of charging speed. Say you drove 80 miles a day and had the occasional day where you drove 120 miles. An 85kWh Model S would still have about 100 miles of usable range left and all you would be doing would be would be adding in a safety buffer for the next day's commute. If your battery was almost empty and you had a 80 mile round trip commute the next day, you would need about 5 to 6 hours with your current charging setup to make sure you had enough range and a bit of a buffer.
4) We have experienced very little battery degradation with our Model S. We have about 35,000 miles on our car now and we usually charge it to between 80% and 90% of maximum capacity. We've taken it on longer road trips for a week at a time and have range charged it (100% capacity) on those occasions and used the Tesla Supercharger network extensively on our road trips. So I believe our car is a good use case in terms of overall battery use. Tesla has changed the calculation algorithms a few times since we first received our car, so it's hard to say what actual degradation has happened. All I can say is that right now my 90% capacity charge is showing about 233 miles of rated range, which translates into about 200 to 220 miles of real world range for us. There are Model S's that have significantly more miles then our car and have reported minimal battery degradation. The battery management technology is more advanced than the Leaf, which is part of the reason for the improved experience of the Model S over time.
The Model S is an entirely different car vs. a Leaf. The Leaf was a car that was definitely a commuter car where I had to be very aware of it's range if I was going to drive beyond 40 miles and I had to be sure to charge it on a daily basis. The Model S is a car that just happens to be electric and I treat it as a regular car for most purposes. I don't have to plug it in every day. Longer-range trips do still require thinking about charging stops, but there is not the day-to-day range concern that can be there with the Leaf.
Perhaps the best way I can explain the Leaf vs. the Model S experience is that when my Leaf lease expired, I went and bought a BMW i3 with range extender. The only reason I did that was because the Model X wasn't available yet. I have a reservation in for the Model X and as soon as that arrives next year, I will be selling my BMW i3 and taking delivery of the Model X!