I agree with the first. There was a time where it was a new idea, but it's so overused. It can still be done well--rarely--so the presence of a "chosen one" isn't enough to turn me off a book by itself. It counts points off, though.
However, I disagree with the second. While I don't want modern names, I think good, old-fashioned medieval or Biblical names are just fine. I wouldn't want to read a fantasy story with someone named Wally, but I have no problem with something like Gabriel or Jonah. (Or, for that matter, William.) To me, the name has to fit the setting, but many real world names fit the traditional fantasy setting just fine.
As to my own pet peeves?
I hate it when a character from the real world who is pulled into a fantasy realm refuses to believe in the reality of it. I mean, sure, a brief period of disbelief is fine. But don't spend chapters or even whole books denying it; even if it's realistic, it's friggin' annoying to read. (Thomas Covenant, I'm looking in your direction...)
I hate it when writers try to make a book feel more fantasy-like by having dialogue written in Ye Old Flowery Englishe. It's fine to have a specific character talk with thees and thous if the author's making that a character trait (like Mandorallen in The Belgariad), but don't make it a standard feature.
I really hate it when authors slap fantasy settings and names on novels that are all about heaving bosoms and pulsing loins, and try to claim it's a fantasy novel rather than a romance novel. While not all their novels are like this, Anne McAffrey and Mercedes Lackey are both guilty of this charge on multiple counts.
Let's see, what else?
If you're going to get your science fiction in my fantasy, tell me on the cover!! I'm not normally a huge fan of sci-fi/fantasy crossover. It can, however, be done well, so I'm not inherently opposed to the idea, if I know about it. Do not try to sneak it in, however, or make it a surprise twist. Ever. Seriously.
The standard fantasy races? Elves, dwarves, halfings/hobbits? If your story doesn't need them, don't include them. Sure, they have a place in fantasy, and some stories are better with them. For the most part, though, if you just need different cultures, use various cultures of humans. When you use other races, there should be a good reason for it. And "It's fantasy, so it should have elves" is not a good reason.
(And now a few peeves that aren't limited to just fantasy...)
If a book is the first of a series, it damn well better say so on the cover!! Nothing makes my arson finger twitch like getting to the end of a book and only then discovering it's book 1 of 27.
Speaking of, tell your stories in a reasonable number of books. The story hasn't yet been written that really needed more than a dozen books to tell, and most don't require half that. And damn it, publish some stories that only require one book! Not everything has to be a series!!
If you're going to publish a book, make sure something bloody well happens. If the plot is in the exact same spot at the end of book 10 that it was at the end of book 9, you're doing something wrong.
Okay, I think I'm done.
For now.