Hi,
As you have eliminated 3 choices in each question, I will explain the remaining two choices.
1. A recent study has found that within the past few years, many doctors had elected
early retirement rather than face the threats of lawsuits and the rising costs of
malpractice insurance.
(A) had elected early retirement rather than face
(B) had elected early retirement instead of facing
(C) have elected retiring early instead of facing
(D) have elected to retire early rather than facing
(E) have elected to retire early rather than face...
Look for parallelism:
E is nothing but have elected to retire early rather than (to) face
B lacks parallelism retirement vs facing
Hence, E

2)A recent study of ancient clay deposits has provided new evidence supporting the
theory of global forest fires ignited by a meteorite impact that contributed to the
extinction of the dinosaurs and many other creatures some 65 million years ago.
(A) supporting the theory of global forest fires ignited by a meteorite impact that
(B) supporting the theory that global forest fires ignited by a meteorite impact
(C) that supports the theory of global forest fires that were ignited by a meteorite
impact and that
(D) in support of the theory that global forest fires were ignited by a meteorite
impact and that
(E) of support for the theory of a meteorite impact that ignited global forest fires
and
D states: the theory that global forest fires were ignited by a meteorite impact,
the theory that contributed to the extinction of the dinosaurs and many other creatures some 65 million years ago.
the second part means the theory itself contributed to the extinction of dinosaurs.

Hence, B