Cat5e is wrapped to a higher standard, and in higher frequencies, carries signals better without degredation, so definetly use it.
For Stranded vs Solid, It all varies.
Solid is best used between two punch down static devices, it can move however it has a chance of breaking when continously bent. Solid cable is best used inside of walls, or other static posisitions with a punch down on both ends.
Stranded by nature bends much easier, it is best used when the cable needs to be continuusly moved, unpluged, repluged, etc. Patch cords are made of stranded material, and in high speed and transfer applications should be kept short due to the inperfections of the cables diameter can cause crosstalk.
Simplified: Hub to wall or computer = stranded, Inside of walls or long runs = solid.