Step 1: Check Statement I.
Shaving soaps do contain glycerol, which keeps the soap from drying out too fast, and rosin, which forms sodium rosinate and boosts lather, true.
Step 2: Check Statement II.
Dishwashing liquid detergents are indeed of the non-ionic type, true.
Step 3: Check Statement III carefully.
Biodegradability actually tracks chain branching, straight unbranched hydrocarbon chains are the ones bacteria break down easily, while branched chains resist microbial attack, so this statement has the logic reversed, false.
Final answer: Option 1, I and II only.