Best solder wire - Sn63Pb37 vs Sn60Pb40 vs ...?
Sn63/Pb37 is better than 60/40 because it is a eutectic alloy. That means it has the lowest melting point of any Sn/Pb alloy, and it solidifies relatively abruptly at one temperature rather than over a range. Generally both are advantages or neutral. 

Combinations with small amounts of (say) gold tend to be for reducing the tendency of solder to dissolve the material (gold in this case). 

Many solders these days avoid the use of lead and are often mostly tin with other materials such as copper, bismuth, silver etc. This is done to reduce the toxicity of electronics that finds its way into the waste stream. In my experience it is worse in every way compared to tin/lead solder except perhaps in applications where high melting temperature is important. 

Flux another matter- there are a number of different types. 

If RoHS compliance (and toxicity) are of no concern, 63/37 Sn/Pb solder with RMA rosin flux is an excellent choice, and is good for high reliability applications. Fine for hand soldering or reflow.  

For production for world markets, it may be necessary to use lead-free solders with more finicky temperature profiles and inferior performance. Sometimes water soluble or no-clean fluxes are acceptable, depending on the product and how much it might affect the process (and possibly the product functionality).