As it is described in chapter one of Five Days at Memorial, Memorial Medical Center was viewed within the community as a trusted institution and a secure haven during hurricanes.

This reputation is established in the opening sentence of chapter one:

For certain New Orleanians, Memorial Medical Center was the place you went to ride out each hurricane that the loop current of the Gulf of Mexico launched like a pinball at the city.

As Sheri Fink goes on to explain, safe haven was extended not only to patients and medical personnel, but also to "kids, parents, grandparents," and various pets for the purpose of waiting out the storms. The hospital seemed to have had a reputation as being near impregnable where hurricanes were concerned.

From here, Sheri Fink describes the hospital's initial, widely publicized founding (as Southern Baptist Hospital) in 1926, which heralded its state-of-the-art status (differentiating itself from the older hospitals of New Orleans). Thus, Fink establishes that Memorial had enjoyed widespread trust and esteem across a span of generations.

The answer to your first question, regarding Baptist's public perception and reputation, can be found in the first few pages that open chapter one. Meanwhile, the answer to your second question will be found further in the chapter. Continue reading, and you'll come across information on those events.