Are Na and Cl ions from NaCl in water separate charged particles?
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).