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The answer to Question 1 is yes, which is precisely Lemma 3.6 in the paper:

Boundary torsion and convex caps of locally convex surfaces, J. Differential Geom., 105 (2017), 427-486.

Although the lemma is stated and proved for $R^3$, the same proof works in $R^n$. The proof also indicates how to give a positive answer to Question 2 as well.