So the first (substantive) issue that comes up in Matthew 10 is the matter of what the names of the apostles were. There are actually many minor issues here, as many of the apostles were known by more than one name, such as Peter a.k.a. Simon, but the SAB points out the most difficult one, this "Lebbaeus, whose surname was Thaddaeus" fellow. Mark's Gospel refers to "Thaddeus" but the two times Luke refers to the apostles by name, he refers to "Judas the brother of James" which is never explained anywhere to be the same person. I think, however, that the fact that a lot of people in the New Testament have more than one name that they go by suggests that it's quite possibly what's going on here. Both Lebbaeus and Thaddaeus may have been nicknames of Judas, as both apparently mean something like "large-hearted".
Should the Gospel be preached to everyone? Well, I don't know for sure, but it seems to me that what is going on in this passage in particular is that at this time, Jesus isn't sending the Gospel outside of Israel, but that doesn't mean that it's never to be the case.
Did Jesus tell his apostles to go barefoot and without a staff? Once again, I have to hand it to the SAB for being so detail-oriented, as I doubt I would have ever noticed this oddity. I think it can be explained, though. In the matter of staves, I think what's being said here is much like what's being said about coats; it's not that they can't have a staff, but that they should refrain from bringing a spare. Going more out on a limb here, as I don't know first century customs regarding footwear, I think a distinction is being made between sandals and heavier footwear, that they can wear the former but not the latter. (I checked, and it is a different word in the Greek, at least.)
At the mention of the infamous ancient city, the SAB asks the question "What was Sodom's sin?" I don't think this question belongs here, but I'll address it nonetheless, as it's a good one. The reason I don't think it belongs is that I don't think that verse 15 or the parallel passage in Luke 10 actually say what Sodom's crime was but only hold the city up as a comparison. (The passage from Jeremiah 23 does likewise.) While homosexuality could possibly have been an issue, I don't see a lot of evidence for it. Yes, Ezekiel says they committed an "abomination", but the Hebrew word for "abomination" that Ezekiel uses shows up over 100 times in the O.T., and only once or twice is it associated with homosexuality, so that doesn't narrow it down much; I'd say it actually is more often associated with improper sacrificial practices than improper sexual practices anyway. No, I think that the passage in Ezekiel lists the real crimes of Sodom, but they are listed in a more general sense rather than a list of specific things.
The SAB puts a whole bunch of its category labels on verse 21, because yeah, it's talking about some pretty awful stuff. While I don't disagree that this is bad, I'm assuming that the SAB is trying to imply that Jesus wants this bad stuff or something, and I don't think that's the case. Intolerance, violence, and injustice are terrible things, but they happen because that's unfortunately just the way people are. The fact that Jesus knew these things would happen doesn't mean that he made them that way. When the next verse says "...he that endureth to the end shall be saved." I think the SAB is reading too much into it to say that this is like instructions on how to be saved. Rather I think it's saying that those who do not endure will not be saved, so take it as an encouragement rather than a sort of rule.
Verse 23 is an odd one, and I suppose it does sound a bit like it's talking about the timing of end of the world. I don't know that it is, though. For one thing, it may be talking about how the apostles in particular will not manage to preach in every city of Israel, although the Gospel may eventually be preached by others nonetheless. For another thing, there may be another meaning to "till the Son of man be come." I think the bigger issue in comparing it to Matthew 24:14 is that that verse says the Gospel would be preached to "all nations" which doesn't imply necessarily "all cities". As it happens, there are Christians in every one of the world's 200-odd countries (it's hard to say how many there are as it changes, and it's hard to say what counts as a country) but certainly not in every city, so the gospel has in a sense come to "all nations" while it very well may have never been preached in numerous cities of Israel.
The SAB asks if we should fear God, a question I addressed at length back here in what I think was one of my better posts. The SAB also asks whether or not Hell exists, which I'm certain I have not addressed. It's actually a difficult question, though, at least in the way that the SAB frames it. That is to say, I'm pretty well certain that the Bible teaches that Hell is a real place, but I will readily admit that it's a place of questionable nature. (I have some reflections on the nature of Hell in my other blog here.) There are those who, no doubt in part due to many of the verses in the second part of the linked page, believe that Hell is a place where the ungodly go after death to be destroyed rather than eternally punished as is the common understanding among mainstream Christians. It may be that some of this is a matter of a sort of poetic use of the term "death" as I've heard many say; when the Bible talks of the "death" of a soul, it may be referring to the eternal torment of Hell as a sort of death. I'm going to set that issue aside as a deeper bit of theology that I feel may be out of my scope and address the "everyone goes to heaven" section of the page (although it may be too deep as well, just shorter).
There is a doctrine that's part of strict Calvinism that suggests that when Jesus died for our sins, he didn't die for the sins of unbelievers, only those who would eventually come to faith. I think that 1John 2:2 suggests that this is not the case, but note that while that verse says that Jesus "is the propitiation...for the sins of the whole world" it does not say that the whole world will go to Heaven. Now when 1Timothy 4:10 says "...specially of those that believe", I think it's approaching the nature of the real truth. The metaphor that I've always gone with is that when Jesus died on the cross, it was like he signed a blank check to pay for the sins of each and every person in the world. The problem of faith is that a blank check is only worth something if you believe in it: everyone has this spiritual blank check, but are you going to sign the back and cash it out? I'm sure it sounds weird, but it works for me, and I hope it clarified a bit where my belief is anyway.
I'm not sure what the SAB is on about with sparrows (marked with the "science" icon), so I'm not sure whether I can address it. As the SAB says, God thinks human beings are worth much more than sparrows, and I'm inclined to agree. If there's scientific evidence that this is not so, I'd probably find it interesting.
I know that the SAB is interpreting verses 32-33 more strictly than I would, and I'm guessing that most Christians would also take this as more of a guideline than a strict rule. In particular, as is pointed out, Peter denied knowing Jesus at the time of his crucifixion, but later in life was a powerful, outspoken evangelist, so I don't think he lost his salvation over it.
Verses 34-37 have a lot of the same issues that verse 21 had, but there may be a few more things to say about these verses. As for whether Jesus is peaceful, it's a bit complicated, and I addressed that issue at the end of this post. I think there is a lot of misunderstanding of the meaning of verse 37 in particular, as it sounds pretty bad (although not as bad as some similar verses). Jesus is simply saying that in the end, our love of God should be our strongest love. Does that mean that we should be cruel to our family in any way? I don't believe so.
Most of the rest of the chapter is flowery poetic talk that I don't know I fully follow the meaning of so well, to be honest, although I can say that I addressed the question of whether there was ever a righteous person in this post.