This is a wonderful but huge series of questions! I'm only going to cherry pick the ones I'm at least somewhat equipped to address. I would encourage anyone with the time or interest to take a look into the timeline, with real primary sources, prepared by the wonderful archivists and staff at the Eisenhower Library ( https://www.eisenhowerlibrary.gov/research/online-documents/civil-rights-little-rock-school-integration-crisis ). Some of the sources I'll reference I've taken directly from the Eisenhower Library site's links, so please feel free to click through there.

One basic note and I suppose, red flag. The staff at the Eisenhower Library like and admire Eisenhower, many of them have incredibly deep and personal knowledge about the man and his family. They care about him - I saw two archivists literally tear up when discussing how painful it was for Dwight and Mamie to lose Ikky in 1921. Ikky Eisenhower was the pet name of Doud Dwight Eisenhower, their first son. This means the staff are often less comfortable discussing some of the less pleasant elements of President Eisenhower's personality, and certainly have an interest in portraying him to the general public in a more dignified, esteemed sense.

The easiest one to address is the "agitators". I think it's important to note what "agitator" means here. This was a mob of white Americans surrounding a high school, beating black Americans - both parents and some reporters, threatening children, and generally spewing vile hatred. If by agitate we mean a violent mob, yes, there were agitators in Little Rock. You'll note that earlier in this speech, Eisenhower points out that "in that city, under the leadership of demagogic extremists, disorderly mobs have deliberately..." etc. The demagogic extremists to whom Eisenhower refers was actually one person in particular - Jimmy Karam. The FBI interviewed Karam about leading the group preventing the Little Rock Nine from entering Central High, you can read the brief interview here -https://digitalcollections.uark.edu/digital/collection/Civilrights/id/1082 . In effect, Karam was willing to speak with the FBI until the agent questioning him directly asked what Karam and the hundreds of people outside were doing, particularly if Karam was in fact continuously going to the school out of concern for his children as he claimed, at which point Karam contacted a lawyer and stopped answering questions. The likely reason for this is that there were several witnesses to the fact that Karam sicced a mob on one of those reporters I mentioned earlier, and then effectively prevented police intervention (imagine the level of violence if a police officer in Little Rock, Arkansas, in 1957 felt compelled to intervene) to the point that one officer threw his badge and baton to the ground and left (See David Nichol's book on the topic for that story). To make another long story short, Karam was a known associate of Governor Faubus.

Eisenhower was an incredibly shrewd individual, but also loved to portray himself as "forced" to act when he acted - whether he did or did not want to act. It's fairly safe to conclude he did not want to act as he did in Arkansas, but also fairly safe to conclude he did feel compelled or forced to act. Eisenhower had tried numerous times indirectly (his preferred method to avoid being forced to directly, as Nixon once described, "shit or get off the pot"), and finally directly on September 14, 1957, to get Governor Faubus to stop impeding desegregation. Basically, Faubus refused to follow the rules and the courts, and (importantly) made America and Eisenhower look bad internationally (also in this speech).

But did Eisenhower mention civil rights in any other context? Yes. He once told Earl Warren he entirely understood the concerns of Southerners, sympathising because all they really wanted was "to see that their sweet little girls are not required to sit in school alongside some big overgrown Negroes.” Quoted in Stanley I. Kutler, “Eisenhower, the Judiciary, and Desegregation: Some Reflections,” in Eisenhower: A Centenary Assessment, page 89.

It, and Eisenhower's perspective, are obviously more complex than that - never take one quote as full evidence of a person's thinking. But generally, Eisenhower hated rocking the boat, he was a middle of the road, postponement-style leader. Civil Rights would be costly in the South, one key reason Eisenhower was so enraged by Faubus's actions was that Eisenhower really did feel compelled to send in the 101, but he worried that Southerners would see this as a second Yankee invasion. Just think about that for awhile and I think you get an idea of how Eisenhower measured Civil Rights.