Muslims would have inherited some version of the stories of Genesis & Exodus from Jews & Christians, but they wouldn't have necessarily needed to be included in the holy book. But, they don't really need to be in the Christian one, either, and there they are.
The Qur'an references, among other stories, Adam and Eve (Q 2:30-32; 38:75; 4:1; 7:189), the Ark and Noah (Q 11:40-48; 54:13; 23:23-30), Abraham (Q 53:36-7; 87:18-9; 2:135; 3:67, 95; 4:125; 6:79, 161; 16:120, 123), Jacob (Q 6:84; 11:71; 12:38; 19:49; 21:72; 29:27; 38:45; 12:93, 96), Isaac (Q 19:49; 37:112-113; 11:71; 14:37-9), Joseph (
Sura 12), Moses Q 2:50-51, 67-71; 7:142; 26:63-6; 7:144-154; 20:85-91; 28:3-43), Aaron (Q 21:48-9; 7:122; 23:45; 37:114-20; 20:70; 26:48; 4:163; 6:84; 25:35; 20:29-36; 26:13; 28:35; 28:34-5), Jonah (Q 21:87-8; 37:139-48), Gog and Magog (Q 21:96-7; 18:94-8), Elijah and Elisha (Q 6:85-86; 37:123-32; 38:48), David and Goliath (Q 2:249-51), and Jesus (Q 19:16-40, 88-95; 43:57-65, 81-2; 23:50; 21:91-93; 42:13-14; 6:83-90; 2:87, 135-141, 252-253; 3:42-64, 81-85; 33:7-8; 4:156-159, 163-165, 171-172; 57:26-27; 66:10-12; 61:6, 14; 5:17-18, 46-47, 72-78, 109-118; 9:30-31).
The above is just a sampling of verses mentioning them. Abraham, for instance, is mentioned in some two hundred and forty-five verses in twenty-five
suwar. Listing all references to everyone is way beyond the scope of this thread.
Do they think of some of those older Hebrew books as nifty secondary books you should also have along with the Quran?
They think of the Qur'an as the last of four holy books revealed by God to the prophets, confirming, completing, and correcting the other three (which are
Tawrat, the Torah,
Zabur, the Psalms, and
Injil, the Gospels). Since the previous messages were all corrupted in some way post-transmission (hence the need for the Qur'an),
I know Islam has these; they can't all be from other stuff like the hadiths, can they?
There are some 500 verses containing legislation, covering things like matters of ritual, alms, property and treatment of orphans, inheritance, usury, consumption of alcohol, marriage, separation, divorce, adultery, theft, and murder), scattered all throughout the Qur'an (almsgiving is mentioned in Q 2:196 and 9:104-5, while adultery is mentioned in Q 4:15-16, for instance)
Also, on the last two bits together: Part of what made me think of this was some quote in which followers are ordered to kill the infidel scum, which made me wonder whether it was just the orders he gave his army once before invading some particular city or more general advice about how to live your life as much as possible wherever you can. In the Bible, I can tell which ones have which contexts to show what such verses really meant.
In the Qur'an, unlike the Bible, the context comes from outside the book, not within it. As a result, what those verses mean depends on which
tafsir you consult. For an academic overview of the various interpretations, I'd recommend
The Islamic Law of War: Justifications and Regulations by Ahmed al-Dawoody and
Striving in the Path of God: Jihad and Martyrdom in Islamic Thought by Asma Afsaruddin.
I've never heard of Islam getting into the talking animals and flying cities and infidel-hunting demons and flaming angels and world-destroying monsters. But, I've also never heard that stuff from Jews, and some of it's in the Old Testament coming from Jewish prophets (just not the parts people talk about much).
It mentions angels (Q 6:8-9; 6:50; 11:12; 11:31; 12:31; 17:95:13; 25:7:13; 32:11), jinn (Q 6:100; 6:128; 18:50; 34:14; 51:56), and things like the splitting of the moon (Q 54:1-2) and Muhammad's miraculous night journey (17:1), or the wall behind which Gog and Magog are imprisoned (see above) to name a few.
I'm sure there are such verses scattered around in there, but the main noteworthy thing about the Biblical Psalms is that such a concentration of them are all in one place, which apparently would not be the case in the Quran.
Correct.
Or any other Jewish lowercase-p "prophet" before Muhammad's time... presumably those people would only be mentioned by the capital-P Prophet to comment on what they said compared with what he said, not to tell irrelevant stories about them.
See above.
But what about stories from his own life?
Unlike the Bible, the Qur'an focuses on Muhammad in the present-tense at the time of his supposed revelation, and contains pretty much nothing about his life and history. That stuff, the
sira, is contained in separate traditions in Islam.
The counterpart would be commentaries about disputes among Muslim sects, but there wouldn't have been much time for those to develop and need to be talked about during Muhammad's life.
Correct for the most part, though Shias do interpret the Qur'an differently and see things that they believe speak directly to their doctrine, things that are quite at odds with the Sunni view.
Any other big subject categories I missed due to lack of Biblical counterpart?
Pretty much any category that's in the Bible is in the Qur'an (or, more specifically, that's in the Old Testament, since Judaism and Islam are a lot closer to each other than either of them are to Christianity).