I currently have 2 week old lettuce and carrots in my fridge in water and they are fine, crisp and fresh.
Lettuce keeps longer in water with a bag over the top. If you catch it before it goes slimy (decomposing), you can recrisp it by putting it in a bowl of ice water (or just water in the fridge). The same applies to any vegetable that is going limp - cauliflower, carrots, parsnips, spring onions, sprouts, potatoes.
Globe artichokes, broccoli and asparagus are also better kept in a glass with a few cm of water. Spring onions too but you have to change the water every day or it will go slimy and oniony. You can also stick spring onions in the garden, they will keep growing! Actually, you can regrow spring onions and lettuce in water or the ground using the leftover end pieces.
The black spots on carrots are from fungus. I wash mine (to get rid of any spores) then store them submerged in water in the fridge. They keep for up to a month that way.
Yellow broccoli is fine, it's just the flowers opening.
I have a special Sistema fruit container for berries that keeps them fresh and mould free for twice as long.