Are Na and Cl ions from NaCl in water separate charged particles? Suppose I pour some sodium chlorine into water. So, what happens is that

$$
\ce{NaCl (s) -&gt; Na+ (aq) + Cl- (aq)}
$$

meaning that the ionic bond between Na and Cl breaks up (correct this far?)

Now, does this mean that the water actually contains separate charged Na and Cl particles? So... since chlorine boils at −34.04 °C according to Wikipedia, why is there then no chlorine gas evaporating? Because it is chlorine ions there, not chlorine atoms?

If I feed electrons some how into the solution, will chlorine gas start forming?

Also, could I use this so that I pour NaCl into water and get Na and Cl ions, and then (since they are separate) add something more to create Na[something] or Cl[something]? Something that would not form if I just mixed NaCl + [something else]. I hope I am not completely lost here..
Most of what you wrote is correct. It is difficult to feed many electrons into the salt water because the Coulomb forces are extremely strong, see 1 and 2, but you can feed electrons in if you drain them out elsewhere so there is no net gain or loss of charge. that is called an electric current.

You can use that to electrolyze salt water, and get chlorine as you surmise; NASA has a nice lab. BTW, you would expect to get Cl gas at the positive electrode, and do get it, but you do not get Na metal at the negative, because hydrogen bubbles out before Na. You can electrolyze molten salt, though, to get a bit of Na (or at least you see it form and then catch fire with a yellow flame).