1. Is it fair to say that options actually have two types of convexity: one due to the payoff structure, working along the underlying’s price range, and the other due to the discounting of that payoff, working along the time to maturity range?
That is correct for interest rate options because the underlying (a bond or swap or term rate) is an extended object. It is discounting-convexity that gives "delta-1" things like bonds and swaps some convexity - for IRIOs, there is this plus the (much larger) convexity from the discontinuity in the payoff.
2. Long convexity is always a positive attribute and, in theory, one should have to pay up for it. How can one price this positive attribute? Case in point the long end of the UK Government bond market. The 2071 gilt has a yield of 4.05, and the 2073 gilt has a yield of 3.92%. An inverted curve of 13 bps. We have two measures, one of roll-up costing one money vs the positive convexity that can potentially contribute positively from volatility.
The theoretical value of the convexity can be computed in a straightforward way given a model for the underlying rates. This is done (for example) in any standard textbook derivation of the CMS convexity adjustment.
3. The Eurodollars curve implies a Conv. Adj., I assume it follows the same rationale as you explained (i.e. Futures vs FRAs), yet given the fact that the US Libor is fading, would it be changed for SOFR? If so, would the so called Conv. Adj change as well?
Yes, it's different for RFR futures; they are based on backward-looking synthetic terms rates.
4. Your scenarios are parallel shifts, right? How about a steepening/flattening scenario?
You are right. My analysis was of the portfolio performance under parallel moves (90%+ of all moves from a PCA perspective). Including more sophisticated curve shifts changes the numbers slightly but not the overall conclusion.
5. Shouldn't the carry on the bond be based on proceeds spread and not the 2 body 1 wings convention?
No, my fly was explicitly constructed with par swaps, NPV = 0 at inception so no "proceeds" involved. But even if I had used cash bonds, the usual assumption is that the bonds will be bought on repo (i.e. on margin essentially) so still no net proceeds generated (the carry is now coupon - repo rate).