Re: Jaime's sword hand
If i recall correctly they are descendants of Garth greenhand (from the female side?). The Gardeners (original rulers i think) intermarrid often with other houses, so a lot of other houses had in truth a better claim or so. They where given the command by Aegon because the Tyrells where the stewards of Higharden. So all they had to do was kneel.
That is like House Poole getting Winterfell and the dominion of the north.
Which is kinda ridiculous, but there you have it.
I stay with my earlier post though when it comes to my view on things. However i should clarify that although i think it'd be awesome if GRRM makes Margaery lose, i actually think it most likel that both of them win their trials.