Answer to question 1: compact subsets of convex surfaces are infinitesimally flexible.

In the Pogorelov book the author proves infinitesimal flexibility of the graph of a convex function over a strictly convex compact subset of the plane. This implies infinitesimal flexibility of any proper compact subset of a strictly convex surface. Indeed, infinitesimal rigidity is projectively invariant (an amazing property given that it is defined in terms of distance). If you send to infinity a plane that cuts the surface in a neighborhood of one point, then the complement of this neighborhood becomes a graph of a convex function.

Basically the idea is that in order to find an infinitesimal bending of a surface with positive Gauss curvature one has to solve an elliptic PDE.

Answer to question 2: closed convex surfaces without flat pieces are infinitesimally rigid.

This is proved in the cited book of Pogorelov (Theorem 4 in Section 4.7). Clearly, any flat piece allows infinitesimal deformations: take any vector field supported inside this piece and orthogonal to its plane.

As to recent research in the field, one may consult a series of surveys by Ivanova-Karatopraklieva and Sabitov. A good textbook on the topic is the first half of volume 5 of Spivak's "Comprehensive introduction...".

One direction in which one can generalize rigidity results for convex surfaces is rigidity of hyperbolic manifolds with convex boundary. (Indeed, rigidity of a convex surface translates as rigidity of a Euclidean ball enclosed by this surface.) In this article:

Schlenker, Jean-Marc, Hyperbolic manifolds with convex boundary, Invent. Math. 163, No. 1, 109-169 (2006). ZBL1091.53019.

infinitesimal rigidity of hyperbolic manifolds with convex boundary is proved (using Pogorelov's theorem, by the way).

Answer to question 1: proper compact subsets of convex surfaces are infinitesimally flexible.

In his book Pogorelov proves infinitesimal flexibility of the graph of a convex function over a strictly convex compact subset of the plane. (Basically the idea is that in order to find an infinitesimal bending of a surface with positive Gauss curvature one has to solve an elliptic PDE.) This implies infinitesimal flexibility of any proper compact subset of a strictly convex surface. Indeed, infinitesimal rigidity is projectively invariant (an amazing property given that the definition is in terms of distance). If you send to infinity a plane that cuts the surface in a neighborhood of one point, then the complement of this neighborhood becomes a graph of a convex function.

Answer to question 2: closed convex surfaces without flat pieces are infinitesimally rigid.

This is proved in the cited book of Pogorelov (Theorem 4 in Section 4.7). Clearly, any flat piece allows infinitesimal deformations: take any vector field supported inside this piece and orthogonal to its plane.

As to recent research in the field, one may consult a series of surveys by Ivanova-Karatopraklieva and Sabitov. A good textbook on the topic is the first half of volume 5 of Spivak's "Comprehensive introduction...".

One direction in which one can generalize rigidity results for convex surfaces is rigidity of hyperbolic manifolds with convex boundary. (Indeed, rigidity of a convex surface translates as rigidity of a Euclidean ball enclosed by this surface.) In this article:

Schlenker, Jean-Marc, Hyperbolic manifolds with convex boundary, Invent. Math. 163, No. 1, 109-169 (2006). ZBL1091.53019.

infinitesimal rigidity of hyperbolic manifolds with convex boundary is proved (using Pogorelov's theorem, by the way).