I remember first starting LotR when I was 11, WoT at 12, after having already read David Eddings' various series. All well before Harry Potter, which as I recall I started reading at 14, borrowing the first books from a classmate and recommending WoT in turn. A Song of Ice and Fire at 14 or 15, Terry Goodkind's books something similar? Still haven't actually read Dune.
Of course, neither me nor my environment are particularly conservative in terms of stressing over what the appropriate age to first encounter sexual or other controversial elements is, so your mileage may vary. I read some other books at similar ages that were certainly worse in that respect (like, say, 1984) - as well as some Harry Potter fan fiction that wasn't remotely as PG-13 as the books themselves.
And while we're on the subject, I also wrestled my way through War and Peace aged... 13ish? and don't recommend the experience. 
Still haven't reread it since, although I dare say I'd appreciate it more now.
View original postDue to the recent screen adaptations of varying quality, on two separate conversations with my brother, who is one of the biggest readers in the family after me, the age at which I first read "Dune" and WoT came up and he was surprised at my age both times. I read Dune when I was 14 and started WoT at 16. He thought both ages were too young. Regarding other stuff in the genre, I read the Hobbit as a preteen, Narnia when I was 10, LotR at 15, the Simarillion at 16/17, Belgariad, Earthsea & Prydain at 14/15. I also, for my sins, read the first Dragonlance trilogy at 16 and the second Shannara series I started at 14 or 15 and finished at 16 (Talismans of Shannara and The Shadow Rising are the first two hardcover novels I ever purchased, though I'm not certain in which order) and the original trilogy at 17. I read a lot of classic sci-fi like Asimov and Campbell, including "Who Goes There" and "I, Robot" at 14-15 because of Dune, WoT came at the tail end of a fantasy kick that was probably sparked by reading a lot of Greek and Roman mythology at 14-15 (I remember 14-15 because I went to a different high school as a Freshman and suddenly had access to a lot of genre books so my memories of reading most of those are in that school & its bus).
I can't comment on Dune, but claiming that 16 is too young for WoT seems pretty absurd to me.
View original postI basically figured these were age appropriate books, because I understood and enjoyed them, even if it took me much longer to be able to analyze the deeper themes and whatnot. But now my niece is a teenager and she's a big fan of Harry Potter and Star Wars and Rick Riordan, and I'm wondering when it's appropriate to introduce these other things to her.
I think you figured right. I'd be more concerned about whether she wouldn't find LotR and WoT too slow-paced / boring, being used to faster-moving YA stuff, than about whether they are age appropriate.
