Can you tell my why Cain is not listed under Adam's genealogy? I have heard it stated that Cain was the offspring of Satan, which leads me to my next question. Does the "serpent" Satan have offspring? Thank you
The first part of your question is simple enough to answer: the genealogies of Genesis establish the line of true (and godly) humanity - for Christ in particular, but also for all of the seed of faith. Note for example that in Genesis 5:3 it says that Adam "had other sons and daughters", but none of these are mentioned either. In fact, the world population exploded after the expulsion from Eden - within Cain's lifetime, he was able to found "a city" (Gen.4:17), indicating that there was quite a large population already (see The Satanic Rebellion: Background to the Tribulation, Part 5 Judgment, Restoration and Replacement III.1: "Satan's antediluvian attack on the purity of the human line [the Nephilim]). Compared to the garden of Eden, the conditions of the post-Edenic world were harsh - but compared to the world after the flood, they were incredibly benign. Under such favorable conditions and with the fertility window for men and women extended 20-30 fold (through their often nearly millennium-long life-spans), mankind's population of the earth of that time was rapid indeed. So it would have been nigh on impossible (not to mention superfluous) to include all of these millions (possible billions) of pre-flood people in Genesis. But it was very important to establish the fact that a line of people has always existed from the beginning of the world since the Lord's re-creation of it who were both genetically pure (with no angelic taint - cf. the Gen.6 link above) and, of equal importance, who were followers of God. For He has never left Himself without witness in any generation (cf. Acts 14:17).
The second part of your question is more difficult to answer. Cain was the son of Adam and Eve. Scripture is very clear on this point and deliberately so, for it is very important for all to see that sin resides in the entire human race and not to blame evil upon a particular "satanic" genetic strain (Gen.4:1). In a spiritual sense, however, Cain was "of the evil one" (1Jn.3:12). Cain, like so many of his kindred in the history of the human race, chose evil deeds (1Jn.3:12; cf. Jude 11), but he did so of his own free will, not because of any predisposed family relationship to the devil. As far as allowing the devil and his demon minions any literal offspring, God has shown Himself largely intolerant. This is because such a half-demonic race would soon overwhelm and destroy true humanity (as almost happened in the days of Noah and occasioned God's destruction of "the world of that time" with the great flood and the subsequent incarceration of the offending demons in the Abyss; see "The Nephilim" in Satanic Rebellion part 5).
There is a very real question, however, given the devil and company's ability to create semi-human seed (provided God will allow it) whether the beast or antichrist will not be just such a half-human, half-diabolic creature. In Genesis 3:15, the Lord tells the serpent "I will put enmity between you and the women, between your seed and Her Seed" - I have capitalized the second "Seed" here, because we know from Paul's explanation of this passage in Galatians that this second Seed is indeed Christ (Gal.3:16-19), the true "Son of Man", the archetypical human being who follows God in every way without sin (Heb.4:15; cf. Heb.2:17). The beast will be a pseudo-Christ, even representing himself as God in rebuilt temple in Jerusalem (2Thes.2:4). 2nd Thessalonians 2:9 also says that the beast's "appearance/arrival" or parousia (a significant word which is also the word used for the "advent" of Christ) is "according to Satan's working (possibly = "fathering"). As a pseudo-Christ, it would make sense that the devil would wish to have his own "pseudo-virgin birth" for the "pseudo-Messiah" he is planning to foist upon the earth. If the devil has any literal offspring, this would be it, not Cain.
Hope this is helpful. Please see also the following links:
Lot, Esau, and Cain: Learning through Negative Examples
Antichrist and his Kingdom: The Beast: All about Antichrist (CT 3B)
The Origin and Fate of the "Giants" of Genesis Chapter Six.
Dinosaurs, the Nephilim, Noah, et al.
Doubts about the Nephilim in Genesis 6
The Nephilim (in BB 2A: Angelology)
Was Canaan cursed?
Why did God order the Canaanites destroyed?
Yours in the true Savior of the world, our Lord Jesus Christ.