Some scholars believe that the ending of Mark is unoriginal due to it appearing to have a different style and the fact that it's missing from some key manuscripts. At the same time, several early Church fathers are believed to have quoted from these passages. What are the primary arguments for and against its authenticity (along with sources, please)?
My understanding is that a strong majority of scholars (including conservative scholars) take the position that the long ending of Mark was not in the original and was not written by the same author as the rest of the text, but nonetheless was added very early on (probably in the early 2nd century). However, the evidence is not as overwhelming as for the Comma Johanneum or the Pericope Adulterae, in part because the long ending of Mark is significantly older than the those two.
The main arguments are as follows. Certainly people dispute some of these arguments, but on the whole each of them is a strong argument, and taking several together gives an even stronger argument.
The textual evidence, which covers the first three arguments, as found in Nestle-Aland is summarized ably at the end of the Wikipedia article under "Summary of manuscript evidence." At any rate all of these are largely undisputed facts, though there are some interesting features of that part of Mark in Sinaiticus and Vaticanus which are worth noting (explained thoroughly with images, though also with a bit of an agenda, here).
A quick summary of the argument from style and vocabulary is given by Bruce Metzger in A Textual Commentary on the Greek New Testament excerpted online here. A much more in depth examination is given in a paper of Travis Williams.
For the last point, you want to first notice many similarities between the long ending and Luke/Acts (as well as a small overlap with the great commission in Matthew), and then you have the trickier point of arguing that it's the long ending taking from Luke and not vice-versa. Wikipedia lists the overlaps (I haven't found a good scholarly resource for this point, though of course you can compare the passages yourself). I haven't found a good reference for arguing in which direction the borrowing is going, but if you want to argue the other way you'd need to explain why Matthew drops almost all of it (despite containing 94% of Mark) and why Luke substantially rearranges it (despite usually following Mark's order reasonably well).
It's worth noting that unlike with say the Comma Johanneum, I don't think it's been conclusively proved that the long ending of Mark is not original. Assuming it was added, it was added earlier than any copies of the text that we still have! All sorts of things are unlikely but possible. Nonetheless it seems the evidence is pretty solidly on the side of inauthenticity.
Short Answer: After weighing all of the evidence (both internal and external), it would seem that Mark 16:9-20 was indeed originally part of Mark's Gospel.
As noted by the OP, the "problem" is whether the end of Mark (16:9-20) was originally part of Mark's Gospel. Essentially the matter comes down to three factors.
1. External evidence
Dr. Constable summarizes the external evidence concerning verses 9-20:
DETAILS AGAINST: absent from א, B, itk, Sinaitic Syriac, ~100 Armenian mss, the two oldest Georgian mss. Some mss that do include it mark it with asterisks or obeli. Clement of Alexandria and Origen show no knowledge of the existence of these verses.3
Despite the absence of these verses in some old manuscripts, and the silence on these verses in the teachings of some church fathers (which could be said for any verse in the New Testament), the fact remains that most old manuscripts do contain these verses, and many church fathers do refer to them. Dr. Constable concludes well that:
Thus, while external evidence in this case is far from conclusive, the evidence does weigh in favor of these verses originally being part of Mark's Gospel.
2. Internal evidence: structural coherence
Dr. Constable mentions an important structural consideration in this debate:
This makes sense in light of Mark's target audience. Mark wrote his Gospel to disciples in Rome who faced a serious decision:
Mark's Gospel served as an exhortation to these Roman disciples to lay down their lives and preach the gospel with boldness. In light of this, it is hard to imagine Mark ending his Gospel "with an example of disciples too fearful and amazed to bear witness to the resurrected Jesus."1 Yet this is exactly what we would have if Mark 16:8 were the end of the Gospel. Thus, structural coherence also weighs in favor of Mark 16:9-20 originally being part of Mark's Gospel.
3. Internal evidence: stylistic clues
Bruce Metzger elaborates on the two main considerations:
Commentator J.R. Dummelow adds a few more:
In favor of Mark 16:9-20 being authentic, Scrivener responds to these arguments with the following dismissal:
Scrivener is highlighting a major weakness in "stylistic clues": they tend to be easily bent to the will of the interpreter! This has been my experience as well, and so I tend to place less weight on such arguments (though they can be very telling in some cases.) But for example:
The arguments against authenticity seem to be rooted in the writings of Eusebius,4 who, ironically, seems to have favored the inclusion of these verses.5 Regardless, both external evidence and internal structural evidence weigh in favor of the authenticity of Mark 16:9-20. Some have argued that internal stylistic evidence points to the contrary, but upon closer examination, these claims seem to be weak at best. Stylistic clues should be considered, but only with the utmost caution, given the nature and track-record of this method of analysis.
Thus, the evidence -- both internal and external -- seems to indicate that Mark 16:9-20 was, in fact, originally part of Mark's Gospel.
1: http://www.soniclight.com/constable/notes/pdf/mark.pdf, pp. 206-207
2: http://www.bible-researcher.com/endmark.html citing A Commentary on the Holy Bible, edited by J.R. Dummelow (New York: MacMillan, 1927), pages 732-33
3: http://www.bible-researcher.com/endmark.html citing Bruce Metzger, A Textual Commentary on the Greek New Testament (Stuttgart, 1971), pages 122-126
4: http://www.bible-researcher.com/endmark.html citing F.H.A. Scrivener, A Plain Introduction to the Criticism of the New Testament, fourth ed. (London: George Bell and Sons, 1894), volume 2, pp. 337-344
5: https://www.apologeticspress.org/apcontent.aspx?category=13&article=704 citing Aland, Kurt and Barbara Aland (1987), The Text of the New Testament (Grand Rapids, MI: Eerdmans), page 287
7: https://www.apologeticspress.org/apcontent.aspx?category=13&article=704 citing Broadus, John A. (1869), “Exegetical Studies,” The Baptist Quarterly, 3:355-362, July.
If you have not read 'The last twelve verses of Mark' from Dean John Burgon, I'm sad to say that you have not fully researched this subject. Please read it, it will honestly vindicate these verses as a true part of the Holy Scriptures as they truly are. This book has not been fully answered by the critics since it publication over a hundred years ago, simply because it takes the right stand for the Word of God. It's too long to put into this little answer box. The book.
Another link to the book in PDF format.
The Last Twelve Verses of the Gospel According to S. Mark Vindicated Against Recent Critical Objectors and Established - John William Burgon
The only two missing these verses are the corrupt א and B.
The CNTTS database shows that the following manuscripts contain
verse 9 A C D05 E07 G011 K017 L019 M021 S U Wsupp Y D Q P Y W 1 2 13 28 33 35 69 118 124 157 346 565 579 700 788 1005 1071 1424 1582 2358 2372
and then jumping to verse 20 A C D05sup E07 G011 H013 K017 L019 M021 S U Wsupp Y D Q Psup Y W 1 2 13 28 33 35 69 118 124 157 346 565 579 700 788 1005 1071 1424 1582 2358
So at least above all of these the testimony of Codices Sinaiticus and Vaticanus are taken to be the true and therefore these God-inspired verses are left out.