1. Both. I prefer to carry, but use my push cart when I have a lot to carry (raingear, cold weather, etc.) or am playing a lot during extremely hot weather.
2. I don't think most courses have a walking path between tees and the fairway. My club does, and all of the walkers really, really appreciate it.
3. It does matter if the grass is wet either early in the morning or due to rain, and it is at least marginally easier to push the cart as well.
4. It would be nice if all courses did this, but it'll never happen so I don't worry about it.
5. This was a decision our super made in conjunction with the head pro and at the urging of some of the walkers. My club really has a very small percentage of walkers vs. riding, so I consider the path to be a very nice gesture from the club to the few of us that walk.