Does the LA-EA2 offer faster focus on NEX-6?

1- Yes. This is based on my prelim tests (indoor and out) on NEX-3. I usually shoot Spot/Local with AF-S and the AF speed is excellent. The only lens that had minor issue was 35/1.8 SAM in low light. But it was still reasonably fast.

2- I would say so. Limited test appears to show good performance. Unfortunately, weather kept me from real world application that I'm planning for, next weekend.

3- No. E-mount lenses cannot work. Strictly A-mount for AF.

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Sony A55, Sony NEX-3, Sony F828