What is the physical size of the Lightning connector?
I haven't seen any precise measurements of the connector itself, but you can see from these iPhone 5 blueprints (which come from Apple for case manufacturers) that the width of the area the connector goes is 9.05mm (including the plastic housing).

I used a hires image from iFixit's iPhone 5 teardown to get an estimate that the metal tip is about 84% of the width of the plastic housing, which works out to about 7.6mm, give or take. Using the same methodology, I'd estimate the height of the metal connector to be about 1.4mm.