The bobber will move two or three times per bite, after which the fish will go away but the line will remain cast. As long as the reel-in occurs before this animation ends, the timing of the click doesn't affect the quality of the catch (the Sonar Potion shows this plainly).
It is possible to get a day's quest fish without having asked the Angler about it, though that would be a lucky event since the player won't know where to fish for it. Taking it to him will then successfully complete the quest.
Time must pass naturally from 04:29-04:30 (new day) before another quest begins.
However, with an Enchanted Sundial, you can get a new quest immediately.
Crate contents are determined upon opening them. Crates caught in normal mode can therefore yield Hardmode items when stockpiled and opened in Hardmode.
Ocean-only catches, including the Capn' Tunabeard quest fish, cannot be obtained from a Hallowed or Corrupted Ocean shoreline, but may be obtainable further out towards the map edge, where Corrupted or Hallowed sand is farther from the water surface. When the Ocean Biome music plays, Ocean fish can be caught.
Possible catches are determined by the biome that the player is currently within, rather than the biome the liquid body is in.
However, the height of the bobber is what determines the height requirements of the catch, not the height of the player.