It's generally 3 AM or 4 AM server time, based on whether Daylight Saving Time is in effect, but the exact time the server resets depends on the server: you can find out the exact time by logging into the game and hovering your cursor over the quest count in the top-left corner of the quest log.

What used to be the minimum required skill to fish in any given area is now the minimum required skill to not catch any junk. Once you meet or exceed this minimum for a zone, whether natively, or through the use of equipment, buffs and consumables, you will no longer catch ANY junk. If you are below this skill level, the chance of catching junk is proportional to just how far below it you are. The further below the threshold, the more junk you'll catch.

The exact formula for the chance of catching a fish when your skill is below the threshold is ((current skill/threshold skill)^2).

For example, the Dalaran Fountain has a no-junk skill of 525. As we can see in @GreatBigFish's answer, this junk threshold is above the (unbuffed) WOTLK maximum skill of 450. Fishing at max skill in Dalaran with no lure would yield a 'fish' roughly 73% of the time. The other 27% of the time, you will catch one of the 13 'junk' items found around the world, all of which vendor for mere copper. To never catch junk in Dalaran, you'd need to use a +75 lure or better at 450 skill.

There is one notable exception to this rule: You will never catch junk in a pool. Fishing from pools gives a 100% catch rate on whatever type of fish spawns in that pool. This has caused many to claim that actively leveling fishing skill is now pointless because it provides little practical benefit - fishing from pools is always more efficient anyway.

For more information, El's Anglin' provides a tremendous resource on the subject of fishing.