1.“2x2 with a square bottom flange with four holes for bolts”
seems to me like a bad design.
With four holes it should be at least 4x4.
2.Personally I don’t like expansion anchors
and find them too weak.
I prefer using drop anchors,
but you do need to be very exact in the depth
of the hole you drill for them and blow out the dust
frequently while you drill.
3.I would use two part epoxy adhesive

http://www.homedepot.com/h_d1/N-5yc1...&storeId=10051


and possibly wait at leas 24 hours
between each hole on one post before drilling the next one.