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.][1]

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.


 


  [1]: http://people.math.gatech.edu/~ghomi/Papers/torsionrq.pdf