Typically, you will want solid cable for stationary runs. This would be your "in-wall" cabling. You use stranded cable for patch cables and cables that will move and bend.
Regarding Cat6 vs Cat6a/e: Get whatever the manufacturer recommends. Most of the ones I've seen (not big into this, though) are designed for Cat5e or Cat6 cable. But, the cost is not that much different so I'd err in favor of Cat6 over Cat5e. I'm not sure if you will see any difference in using Cat6a/e (i.e. 500Mhz).
It is quite difficult to terminate Cat6 to spec if you are a beginner. So, I'd recommend getting a pre-made patch cable of 100ft. if that would work in your situation.
Regarding 250 vs 500/550Mhz: Generally, the higher the number the higher max. bandwidth at distance you will achieve. As mentioned already, Cat6 is rated at 250Mhz and Cat6a is 500/550Mhz. Again, generally, Cat6a is designed to support 10G Ethernet at a greater distance than Cat6.
Hope that helps.