Mounting / Support requirements for an oversized book case
The angle brackets alone should be adequate if two things are true: Your case is well-built, and it's fastened securely to the wall. The spindles will do little to support the unit should they be accidentally knocked out by children, pets, or you. They shouldn't be considered a critical support component, in my opinion.
No, don't use nails. They don't have the pullout strength of screws and aren't removable (practically speaking). Use large wood screws (#14 or so), or lag screws (1/4" to 5/16" or so if the bracket holes allow). The key is penetration into the framing lumber. I'd want 2" of depth into the lumber itself. Pilot properly for maximum strength (slightly smaller than the screw shank). Also a factor is the number of screws per bracket. I'd want at least two. Also a factor is mounting of the cabinet to the wall. If you have a stout back panel, or a set of rails through which you can fasten, decorative or hidden screws into the framing will take much of the load off the brackets.
I addressed this earlier. Any support legs must have adequate diagonal bracing to prevent knockout. This means that the effort and expense may make them impractical. In any case, 1" is probably too small. Look to heavier oak stair balusters in the 1-1/2" or 2" size, with the milled dimensions being slightly smaller.