Downed power lines are dangerous because they may still be live.
A downed power line (or any other ground fault) can be deadly even if you do not touch the line itself, due to the phenomenon of step-potential and earth-potential rise. Check this Wikipedia reference for more info on step-potential:
To answer your second question: Power grids do use electrical protection systems to stop the flow of current in event of a fault. However, for an LV line (110Vac in America, ~240Vac most other places) you may find the nearest upstream protection device is just a fuse. For MV lines (1000Vac+) the protection is likely to be a device called an auto-recloser or sectionaliser, which are more sophisticated than fuses but still cannot be relied on to make a downed line safe.
An example of why a downed line might remain live is if distributed generation (e.g. solar panels or a local wind turbine) have been connected to the grid downstream from the nearest protection device. The distributed generation could potentially continue to feed the downed line even after the protection system has operated. (If the DG has been well designed and implemented it should have its own protection system to prevent this, but it remains a possibility.)