Rosen core is for electronics, Acid core is for everything else, Solid core can be used for either as long as you use the proper flux (Rosen or Acid). The purpose of the flux is to remove any impurities from the surface of the metal BEFORE the solder melts and binds to the surface. It does this by having a lower melting point than the solder.
Solder does not go bad over time (it's just lead and tin) however the fluz in the core can dry out and become ineffective, in this case you can apply additional flux from a bottle or a flux pen.
To prevent the flux in your solder (any type) from drying out simply squeeze the end of the solder closed with a pair of plyers before storing it.
All solder joints should be properly cleaned of any flux residue after soldering. Use baking soda and water for acid flux and alcohol (Everclear works very very very well) for resin flux.
All types of flux if left on the joint can cause corrosion over time, it's best to clean it now than pay for it later.
One quick note about Everclear, it's 98% pure. Cleaner than de-natured and cleans better. You can use it anywhere you would use Isopropyl or Denatured. It's a great cleaning solvent, leaves no residue, you can even use it to clean audio/video heads. It can be used to clean that cut on your finger. And if all else fails you can even use it for "medicinal purposes" after a crash
I look at it this way, if it's good enough for NASA to use as a cleaning solvent, it's good enough for me!