Does the point at which salt is added to a dish affect how the flavor is changed?
Yes, the flavor is definitely affected. 

First, there is the point of solubility. Salt doesn't dissolve all that easily in water, and unlike many other crystals, its solubility doesn't improve with heat. So you have to either stir a lot, or let the dish to sit for a longer time after the salt is added, or both. Else you end up with uneven salting in liquid dishes like soups. Thicker things like sauces are very marginal, and if you add salt to cooked dry stuff (steaks, casseroles and whatnot), you're guaranteed to end up with clumps. 

But even if you succeed in getting your salt evenly distributed, the flavor is different. The liquid components may be OK, but anything solid (pasta, meat, vegetables, whatever) will not be penetrated by the salt. So if you salt your dish to the point where the liquid tastes normal, you get bland pieces swimming in an OK sauce or broth. At that point, most people just automatically think that it is still bland, and continue to salt, resulting in oversalted broth/sauce with bland veggies, which in my opinion is even worse. 

Third, the salt isn't used only to add taste. A saline solution's chemical properties are quite different from those of pure (or tap) water. Marinading with salt (with or without liquid) changes the texture of steaks (but does so rather slowly). Vegetables cook firmer in salty water, instead of getting mushy. And then there are all the amazing things it does to grains (because it affects both starches and gluten). In short, don't bake a yeast dough without a pinch of salt. 

Sadly, all this means that there is no way for both of you to eat what you want/should. My advice is that you should definitely salt the food even for the low-sodium eater at the proper cooking time, you just should use a much smaller amount - even a gram of salt per liter of water can be useful for the chemical changes you want, and shouldn't add too much of a sodium load to a normal sized portion. As for bringing your portion to your own taste, you will have to put up with the worse saltening. Just take care not to oversalt as a reaction, it doesn't really help.