So in chapter 16, God says it's time to move on, and look for the new king. Samuel is nervous, because he thinks Saul will attack him if he tries to choose a new king, which was probably an accurate assessment.
God tells Samuel to go to Bethlehem, and there give a sacrifice, inviting Jesse and his family because one of Jesse's sons is destined to be the new king. Everything appears to be going according to plan until God tells Samuel that none of the sons of Jesse present are the right one. Samuel asks Jesse if he has another son, that son being David, of course.
How many sons does Jesse have? Honestly I don't know. There's definitely some confusion between here and 1Chronicles, and the nature of the resolution is not obvious. There are a few possibilities. One, that completely erases the contradiction, but at the cost of also garbling the intelligibility a bit, is that technically, Jesse did have seven sons pass before Samuel in total. This makes verse ten very badly worded, but it's not entirely implausible. Another possibility is that Jesse had eight sons at the time of this event, but one of them died before having a child, possibly in the fighting against the Philistines. Thus the later claim that Jesse had only seven sons means he had only seven surviving sons. Actually, now that I put it in words that way, another unlikely but possible scenario comes to mind: "sons" could be a sort of mistranslation of "children", and Jesse may have come to the sacrifice with six sons and a daughter. Honestly, the most likely answer is that the latter verse was a mistake by someone who read this story and didn't do the math, but I have to consider possibilities. (Edited to add: The very next chapter says plainly that Jesse had eight sons. The SAB might want to add that verse to the contradiction page for added clarity.)
I'm not sure why the SAB marks the anointing of David as violent and unjust. Maybe there are aspects of David's kingship that the SAB sees as those things, but the anointing itself is hardly violent.
It is definitely a very strange thing to hear that God would have sent "an evil spirit" to Saul to trouble him, but there are ways to look at this event that may or may not seem acceptable depending on your theological viewpoint. Taking it from a purely straightforward point of view, that is, to say that God literally sent some sort of evil spirit to Saul, might be acceptable to some who don't see a conflict with God enacting evil through a proxy, which is something like what some Jews believe. I tend myself to think this is figurative language, and it's not so much something like a demonic presence, but simply a "spirit" in the same sense that one would talk of "Christmas spirit" if one is not Charles Dickens. In that case, whether it was literally God actively tormenting Saul or rather that Saul felt tormented in knowing he was to lose the kingdom and the favor of God is up to you to decide; in some sense, I suppose it boils down to much of the same effect.
Now the end of the chapter has Saul meeting David for apparently the first time. David is hired on to come to Saul's court and play music for him to make him feel better, and David eventually comes to be Saul's armorbearer. This seems to be a contradiction with the story in the following chapter, which also appears to be a story of Saul meeting David for the first time. In my opinion, it is the latter story that represents the first meeting of Saul and David, and it is through that latter (in the book) meeting that David becomes well-known enough for Saul's courtiers to recommend him in this instance. This story would then not be in chronological order, but placed here for thematic purposes to follow the story of Saul's "troubled" feelings in losing the throne.