Prone is a condition that can apply to all creatures regardless of circumstance or terrain.
So we're definitely looking at a character going prone in the water being a thing.
How that is ruled is somewhat up to the DM since you have the freedom to model prone however you want if the basic circumstances don't fit (ie lying flat on the ground). All normal conditions of prone are in effect (move action to stand up/go back to normal), combat advantage against melee, +2 defense against ranged etc.
Based on the fact that creatures that are flying or climbing fall when they are knocked or drop prone, I'm going to say that a logical conclusion is that prone creatures who are swimming are going to sink (probably just 1 square as with failing a check by 5+). This isn't in the rules, but it makes sense.
As far as whether or not a creature can crawl, you may attempt any movement method in the water, it requires an athletics check to do so if it isn't a swim speed (or presumably a teleport?). A crawl would have a movement speed of 2 and would end up good for 1 square of movement.
As far as being unconscious under the water, you would simply begin sinking. Check out the endurance skill for guidelines on drowning. Generally you get 3 minutes under water before you need to make checks, so that hopefully enough time that you can healed up and back in the action. Though more details here should definitely be it's own question after you've read that section.