370C is beyond the melting point of most types of solder so it should be ok for simple solder jobs but an iron with an adjustable temperature-controlled tip would be suitable for a wider range of solder work. Without being able to lower the temperature you run the risk of damaging/burning small circuit board tracks/pads and any sensitive components. The temperature should ideally be just enough to melt the solder joint quickly (like a second or two) and no more. When the iron is too hot, or you need to hold it on the joint for several seconds because the iron isn't powerful enough, the heat can cause tracks and pads to lift or break away.
Similarly, 60W will be ok for most solder jobs. It'll be sufficient for soldering most larger components, such as power sockets/jacks, but you might still struggle on some solder pins that dissipate a lot of heat, such as those connected to heatsinks or very large ground-planes. The iron's wattage doesn't define the tip temperature but rather its ability to maintain the tip temperature. So if you take a feeble 30W iron, for example, and try to unsolder a large heatsink, the tip temperature will plummet when you bring it to the solder joint, making it impossible to melt enough of the solder, regardless of what the initial tip temperature was. Of course if you hold the iron on for long enough, you might eventually get the joint hot enough, but by then the heat will have spread to the circuit board and surrounding components, potentially risking damage.
