Most of the objections that the SAB gives for this chapter are pretty trivial, in my opinion. There's a lot going on here that's confusing and sometimes seems harsh, and indeed, it is to an extent, but taking the long view, I think it all makes sense.
Like the previously-discussed passages in which God "appears" to people, I think saying that God "visited" Sarah is really a figure of speech. Really, God is with Sarah all the time, as He's with everyone, but in this case, what there was was a visitation of God's power upon Sarah's post-menopausal reproductive system. As the SAB points out, Abraham doesn't seem to need any help in that area, as he has several children after Sarah dies years later. The SAB points out that "God-assisted conceptions never result in daughters." which is technically not true, as theologically Christians and Jews believe that all conceptions are God-assisted in a sense, but the point is clear enough. I'm not sure why this needs to be pointed out, unless the implication is that there's some sexism involved. From one point of view, it's often the mothers that seem to care the most about infertility, and a point could be made that these special conceptions are a special blessing to the women in particular, but I can certainly understand that that might not convince everyone. (Remember that Abraham was perfectly happy to have Ishmael be his heir!) If Steve Wells is trying to make a different point, perhaps he can leave a comment.
So Sarah has a child, and he is named Isaac, a word often translated as "laughter". There's a lot of interesting usages of this word surrounding the life of Isaac and his parents: Abraham laughs the first time he's told Isaac is going to be born, and so does Sarah. The word can also mean "mocking", "playing" or "messing around". It's the word used of Ishmael's actions towards Isaac in verse nine, and even Issac's actions with his wife in chapter 26 (probably some sort of "heavy petting" as they say). Because of the ambiguity/flexibility of the word, verse six may possibly mean Sarah's saying something like "God's making fun of me!" but certainly in a positive way. But in the midst of this mirth, Sarah gets angry when she sees Ishmael doing whatever he's doing to Isaac. (Although "mocking" is most likely the best translation, some have suggested due to the word's usage elsewhere that he may have simply been laughing in a friendly manner, and Sarah somehow couldn't stand to see this illegitimate child happy, or the possible other extreme that Sarah caught Ishmael molesting his younger brother. The latter would probably give very good justification for Sarah's anger in this story, but there's not a real strong basis for this speculation as far as I know.) Sarah insists Abraham send Ishmael and Hagar away, and whether or not she's at all justified in her anger, God seems to feel that it's a good idea, and tells Abraham to do it.
The SAB claims that Abraham sent them "out into the wilderness to die" but I think there's an important point missing from such a reading of the story. Even if it wasn't reiterated by God here (which it is), Abraham no doubt remembers that God already promised that Ishmael would come to eventual prosperity. No, the point wasn't to have Ishmael dead, but to have him come into his own someplace a bit farther away from his half-brother Isaac. Note that this is a repeated theme throughout Genesis, that two close family members are too prosperous to be able to coexist in the same general area, and they have to part ways. This is just preemptive in Ishmael's case. Abraham gives them food and water and sends them off.
The SAB questions Ishmael's age at the time he is sent off with his mother, as the general context of the story suggests he's in his mid-teens, but some of the particular wording in this passage suggests a much younger child. First of all, the wording of the translation in verse 14, while technically following the wording of the Hebrew, unfortunately doesn't quite capture the spirit of the grammar, unfortunately. Note that the phrase "and the child" comes after the phrase "on her shoulder", and the implication is that she's carrying the bread and water, but not her son. Grammatically, "the child" follows the verb "gave", but not the subclause. Admittedly, it's not real clear, but apparently, this is just a difficult to translate passage. I don't think verse 15 really poses a problem, as one could certainly even take a full-grown adult and shove them under the shade of a bush. Verse 18, on the other hand is much more awkward phrase, as the KJV translation makes Ishmael sound like a tiny infant; many other translations use "by the hand" or "with your hand" which make more sense for a teenager. (I have a whole spiel about translation of prepositions that I usually share in the context of chapter one, and I can't believe I skipped over it! I even promised I'd include it in one of my early posts. I'm going to add it to the comments here later today.)
Back to Abraham, we see that he is approached by King Abimelech and Captain Phichol, who tell him that they respect Abraham's standing before God, and they'd like to make a pact with him that they will all do their best to get along, and hopefully they will all prosper. Note that we have seen the name Abimelech before, and we'll see it again, once with the name Phicol. While some instances of this may be the same person, it's very likely that "Abimelech" is more like a royal title than a proper name in the usual sense (it means "the king is my father") and Phicol (meaning "strong") may also be a title for the head of the army of Gerar. When Isaac runs into these guys, it's about 40 years later, and he has a similar sort of experience to his dad's first run-in with them.
Abraham makes a pact with them, which takes a bit of working out of some details concerning a well and some sheep. Swearing, making covenants, and taking oaths are good things to do, but Jesus gives an important caveat, which is echoed by James: if you understand the cultural and scriptural context for the two quoted verses, the point of the verses is not that someone should abstain from swearing (although it is true that some Christians, to be on the safe side, have "sworn off" swearing, if you will) but that swearing should not be an excuse for hypocrisy. When you were a kid, did you ever hear a kid say (or say yourself) "I swear on a stack of Bibles!" Why should swearing on a stack of Bibles be better than swearing on one Bible? Why should swearing on one Bible be better than just swearing? Why should swearing be better than just saying something, and people know you to be of such good character that they know you mean what you say at all times? Paraphrasing Jesus in Matt. 5:37, "Just say what you mean all the time, and swearing should serve no purpose."
The chapter ends with a couple of questions by the SAB concerning some geography. I'm going to answer the questions together, as they are related in a way. This story is supposedly the time Beersheba is given its name, and although another story is given later, the SAB apparently missed the fact that earlier, Hagar and Ishmael were sent away to Beersheba, which seems like it would be a clear anachronism. The thing is, while the area and the well in that story were named later, the author is writing after the fact, and calls the place by the name it was known by in the time of the writer. Beersheba was still called that 1,000 years later, and as far as I know, is still called that today. My point in answering this question that was not asked (but maybe should have been?) is that when the Bible here refers to "the land of the Philistines" hundreds of years before Philistines, it may simply mean that it was the land that eventually would belong to the Philistines. Also, as with so many other names, the meaning has a vagueness that means it may mean nothing so particular at all, as the Hebrew word for "Philistines" means "immigrants". As for the naming of Beersheba being covered twice, and seemingly thus contradicting, there are many interesting things going on in the Hebrew. There's a bit of wordplay involved here, as they make an oath (shev'uah) over seven (sheb'a) ewe lambs. In Isaac's story, there's no sense of "seven", but only an oath involved. It may be that while Abraham called the place Beersheba, the name didn't quite stick until Isaac's time. It may also be that Isaac's well was not just his father's well redug, but a new well in the same general area. Since Abraham and his son keep having to dig new wells here and make new covenants, the name just eventually comes into common usage, I guess, and the name sticks for over 3,000 years.