1) In general, there's no hard rules - it's up to the DM. Certain types of curses may not be hidden. Intermittent functioning, requirements, and drawbacks might not be secrets difficult to discover. The "Analyze Dweomer" spell might well show them, or other curse effects. Other divinations might make curses more obvious - a well worded "Augury" might reveal if trying on a ring is a bad idea, etc.
2) There's no "standard" way to permanently remove a curse from an item. The DMG notes that many so called "cursed" items are not so much cursed as they are malfunctioning or otherwise magically malformed. For such items, perhaps a partical re-crafting is in order. Again, this is up to the DM. Maybe it requires the blessing of the world's deity of magic, to be earned by a quest. Maybe it requires a Wish spell. Maybe it cannot be done at all. Use your imagination, here.