Chapter 18 is the prelude to the destruction of Sodom and Gomorrah in Chapter 19, and some elements of this story make more sense when you put the two chapters together, in my opinion. First of all, the SAB notes that "Abraham feeds God and three angels." Various religious traditions have different understandings of this story, but I'm not aware of any that have that particular understanding. If I'm not mistaken, the Jewish tradition is that God is not physically present at this meal, but only these three angels. Christian tradition states that either God is one of the three or, being a Trinity, God is all three at once. In any case, something one can see in the next chapter is that while three men come to Abraham, only two men/angels arrive at Sodom. I think the Jewish tradition might be that the third went to Gomorrah, which makes some sense, as it seems that if there's a need to send physical witnesses of some sort into the city before judging it, then both cities needed a visitation. Something odd about this story overall is that while both the cities are destroyed for their wickedness, we really don't ever hear any details about Gomorrah's issues.
While I have already addressed the question of whether or not God can be seen, something interesting to note about this story is that there seems to be little indication that Abraham realizes that these men are anything special at first. The wording of his greeting to them is different than the way he addresses God in chapter 15, but then again, he also speaks to God differently in verse 27 onward, when it seems pretty clear that he does know who he's talking to, so it's hard to say. (In the KJV, a word translated all-caps to "LORD" or "GOD" is a translation of the Hebrew name for God. At current, the SAB has the capitalization wrong for this chapter, so you can see the correct version here.) In any case, I think the real point here is a contrast in styles of hospitality between Abraham and the Sodomites. It's understood that proper etiquette in those days was that when you had a guest in your home, you would almost go so far as to act like their slave to show them proper hospitality, which may explain some of the bizarre actions of Lot in the next chapter. Abraham suggests humbly that he will get the men "a morsel of bread" and when they agree, he runs off and has a banquet prepared for them.
(Interestingly, the meal includes both meat and milk, a Jewish dietary no-no; while the dietary laws have not been handed down yet, it's fascinating to note it. Jewish tradition is that Abraham, despite living before the law, kept kosher, and many theorize that while he served meat and milk at the same meal, it was not part of the same dish, and the angels probably only ate one or the other.)
So the men ask where Sarah is. Once again, this is not a case of God failing to be omniscient, but merely a conversation-starter. While it may be possible that one of the angels asked, and he himself did not know the answer, it's not like they called her out to talk with her, they're drawing the subject to this message of her upcoming pregnancy. When Sarah overhears the message, she laughs, because she thinks it's silly since she's well past menopause, and perhaps isn't even having sex with her husband anymore. God hears Sarah's thoughts, and tells Abraham, asking "Is anything too hard for the Lord?" The SAB points out a few verses that name some things that are apparently too hard for the Lord, and while I'm tempted to simply answer the question "No," and move on, addressing those verses when I get to them, it's an important enough question to give time to here.
The three verses that are quoted on the page each have their reasons, and let me address the last one first: Yes, God cannot lie. This is not to say that God does not have the physical ability to lie; any being that has the ability to give information could conceivably give false information. However, God, by His nature, is unable to do anything but tell the truth. As for the other two verses, the answer is probably similar for both of them. In Mark 6, if you look at context, you see that the problem there is lack of faith. Some people have supposed that God is like a big engine that uses faith as a fuel, but that's not why faith is important. Note that just about everywhere else Jesus went, big crowds gathered around Him, eager for His wisdom, his forgiveness, and His miracles. Here, nobody shows up. What is likely happening is that Jesus is interested in doing miracles for people, but only for people who really want to see miracles, and are putting their full trust in Him. Jesus wasn't able to do much in that town because nobody wanted Him to, and so there was no point for it. The similarity in the Judges passage is that the people of Israel were fighting for God to take the land, but it probably was the case that when they got to whatever valley this was, they saw all the chariots and lost hope, never taking hold of God's promise that He would make them victorious. Note that it doesn't say they lost against the chariots, only that they couldn't do it.
So before they leave, God says, "Shall I hide from Abraham that thing which I do...?" God's not "worried that Abraham might stop Him", He's asking a rhetorical question; it's His way of saying, "Y'know what? I ought to let Abraham in on what's about to go down here." So He explains it. When He says He "will go down now, and see" it's not an instance of Him being unable to see from Heaven, but something else. There's a pattern that recurs in the Bible fairly often that God always takes some concrete action before judging, in order to give a last chance. In this case, it occurred to me that God might be giving these angels the final say on the matter in a way, telling them, "I've decided that there's not much worth saving in these cities, so go down there and take a look; I've given you authority to do with them as you see fit." I think a more orthodox approach would probably be that the angels coming to visit is a final test to see how the people respond. More on this next chapter, I think.
Abraham, upon hearing the news, asks, as the SAB puts it "two good questions" about the destruction of S & G. These two questions are the beginning of a series of similar questions, of the form, "Will you destroy the city if there are X righteous people there?" Most people interpret this as Abraham bargaining with God, but the concept bothers me to some extent. It seems to me personally that a possibility is Abraham is trying to understand both the nature of God's justice and how wicked the city is. "Surely you wouldn't destroy a city if it had at least thirty good people in it, right?" "That's correct, Abraham." Maybe Abraham is indeed hoping if he gets the number low, God will change his mind, but I think Abraham is possibly thinking in the end, "Man, there's not even ten righteous people living there? What's my nephew doing living in a place like that?"