Handicap vs SG: Does it mean that how the calculation of round handicap for a specific category (e.g. Short Game) does not have additive property as stroke gained has ?

Exactly. That’s why I usually prefer strokes gained per round, because you don’t need to spend as much time on improving a skill that has little impact on your score. However, if you just want to measure skill in a given area (independent of the number of shots taken per round), then strokes gained per shot is more appropriate.

Q: Am I correct in that this player is losing 0.2 strokes per shot in each category compared to a Scratch golfer?

A: Yes, that’s correct. The ‘handicaps’ are based on SG per shot.
I often look at the other report that has strokes gained per round. In that report, the SG Total per round is the sum of SG per round in driving, approach, short game and putting.