Gearing questions aside, yes $50 is a good deal and yes, a 6800 bottom bracket will work with a 6700 crank.

However, changing to a 10-speed cassette will; also require changing to 10-speed shifters/brifters and that's where it gets expensive. You could keep your current 9-speed cassette and shifters and just use a 10-speed chain or possibly even keep your 9-speed chain. Your current front derailleur can probably be used by resetting the low limit screw to lock out the granny position.