Diablo III – Is It Safe to Leave Character Idling?

This is currently not possible.

However, I think it's safe to assume that Blizzard will add this as a paid service at some point just like they did in WoW.

There is only so much lore in the game. I suspect this is to preserve the focus on the action and adventure, and to allow the player to identify with the hero more, making it more 1st-person than 3d-person. However, Sanctuary is a world rich in storytelling.

For lore on the heroes, I recommend reading the 5 short stories released by Blizzard shortly before the game: The Heroes of Sanctuary. I have read a few of them and they are excellent, not just hack promotional literature, but vivid character pieces in my opinion.

Second, there are novels set in the world of Sanctuary: List and Chronology of Novels. The novels Moon of the Spider and The Order take place in between D2 and D3, and The Order features a Monk character as well. Here is an interview with the author of The Order which goes into some details.

Then there is of course The Book of Cain, which is a book posed as written by Deckard Cain. It goes into lore, histories, angels and demons, and more. It also covers events in between D2 and D3.

There is also the Sword of Justice comic series, an ongoing miniseries that takes place right before D3, and involves a Wizard character.

There may be other lore pieces that Blizzard has produced and released as part of the game's marketing and media, but I think the above covers most of the official lore releases.