For shoes I prefer the New Balance Minimus 10: New Balance Minimus
I have a couple pairs of Vibrams but I don't particularly feel comfortable at Crossfit in them. I've seen others in the Minimus as well. The issue with running shoes is usually that the soles are squishy and that's not really ideal when you are talking about the Olympic lifts or even heavier lifting in general. What I discovered when I transitioned to the Minimus was that my flexibility intially sucked, but after wearing them for a few months it's unbelievable how much more depth I can get in squats, etc. They are a great all around shoe for lifting, plyo, anything else you will encounter in Crossfit.
That rope looks exactly like the cable rope I just bought from Rogue. It's a great rope to start getting double unders on. It is adjustable. There is a screw and nut on the end of the rope that you can use to adjust the length. The bonus is you can get replacement cables for it pretty cheaply if you jack it up.
As for hand strength, I myself have been struggling with that for some time. Being in Crossfit for a year now has helped tremendously. My grip strength improved vastly mostly, I think, due to farmer's carries with "heavy" kettlebells (35 lbs for me is pretty heavy but I've carried heavier for farmer's walks). There are grip tools you can buy, but I have found that to be unnecessary with more time spent at Crossfit.
~All luck is earned in the end.~