60/40 solder is fine for most all electronics, with the main thing making sure that the parts don't move as the solder cools. 63/37 is the best as it has no plastic reason but probably isn't worth purchasing for this one job that's already done anyway.
The size of the iron and the diameter of the solder are based on the size of the parts being soldered--the metal you are soldering and the solder itself are heat sinks to the solder iron.
If the part is already fixed, you should be okay. Putting down flux before hand does make the parts clean and improve the connections.
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