Board on top of joists in attic. Is it structural

Can I screw the blocking instead of nailing it as I'm a little worried the nailing might cause cracking in the plaster ceiling (with wood lath) below?

Absolutely.

Is it okay to sister a 2x6 board to the current 2x6 board along the front and back perimeter of the finished area to support the ends of the new subfloor since I'll be cutting the current subfloor back to the edge of the current 2x6's? Can I use screws instead of nails?

Again, absolutely. Definitely makes things easier.

Should I replace the loose fill insulation with unfaced batts and fill in any small areas with the old loose fill?

Not necessary if over conditioned space, but would not be a problem.

Does all of this sound kosher / is there anything else I should be considering?

The joist are braced by the lath and plaster below on the bottom , and the subfloor on the top so no bridging will be needed.

Toenailing into the top of the beam would probably be adequate, if you use slight care until you get the floor fastened to the top of the joists. For a more robust attachement, a 2x6 or even a 1x6 attached to the ends of the joists (nailed into the ends) such that it overlaps the side of the binders with the extra inch will be quite robust. Serves the function of a "rim-joist." Some blocking between the joists at mid-span or third-points would likely be overkill, but is another approach to resist overturning.

I would recommend the above approach.

"hurricane ties" are a metal bracket that may be handier (and may cost less) than the two you mention. They nail to the face of the joist and the face of the binder. But probably overkill for an interior light storage floor. You should be able to install the shortest version of them by simply pushing the insulation down while you nail, and letting it spring back up once installed.