I expect that u can. However, I would think that you may want to make sure that it stays secure when you turn the water on so that it works correctly.
A leak
$500
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No, you don't have to replace your sink, you just have to replace the fixture's seals. That will stop the water from leaking.
yes
Replace the faucet.
There are possibly clips under the sink that need tightening. If not, lift the sink up and put a bead of silicone caulking under the rim and set the sink in it. Once it dries, the sink can't move.
Replacing a bathroom sink is fairly easy and definitely worth it for you to do for yourself. You can find instructions on how to replace it here: http://www.ehow.com/how_2099972_replace-bathroom-sink.html
You replace them with a non ferrous metal such as copper Type L or Brass piping schedule 40
An average bathroom sink is 6-7 litres
This depends on the size, brand and type of bathroom sink you are looking for. Another factor to take note of is that every bathroom sink installation has a different waste installation which wont be included in the diagram.
Bathroom sink use 5 liters of water.