It is really funny to see how the TTers try to discourage the OP to travel by car without answering any of his original questions.
1) There is nothing ilegal to sleep in your car but it can be suspicious and it could be ilegal at some specific places in each country. But with common sense you should have not much problems
2) Possible. Also at gas stations, especially along the highways you can find showers.
A car means always much more freedom and is essential to drive along the countryside and explore smaller towns. These places are often very difficult to explore by public transport. In the bigger european cities a car is usually not a good idea, but you can park and sleep in the outskirts and take public transport to get into the city center.
Safety isn´t much a problem except in some parts in Italy.