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.

Sure it does. Try cooking pasta in unsalted water, and then adding salt afterward... it's quite a different experience. If you add salt early, the salt cooks into the food. If you add it at the table, or even later in the cooking process, the salt doesn't permeate the food to the same degree (or at all).

Taking the pasta exercise one more step, if you add cheese or a slightly salty sauce, the pasta might taste even more bland due to the contrast.

A cooking situation that is analogous to this is brown sugar on oatmeal. If you devolve and mix the sugar into the oatmeal it will taste much less sweet than a sprinkling of brown sugar that melts on top. The sugar because it's suspended in the oatmeal and doesn't touch the sensors on the tongue when it's mixed in doesn't get noticed. However, when sprinkled on top the sugar touches the tongue and the one bite of food tastes very sweet even though there is less sugar than when mixed in.

If you salt water before boiling pasta in it, the water and salt will be even absorbed into the pasta so while it is being eaten and tasted there is an even distribution of salt hitting the sensors on the tongue. So, yes, there should be a distinction in flavor depending how and when the salt is added based on how much salt is suspended or absorbed into the food item being cooked.

It's not so much that the salt is transformed as much as how the sodium is delivered to the tongue with each bite.

If one omits adding a small amount of salt at the beginning of cooking, much much more salt has to be added at the end to achieve the same level of saltiness.

Or this has been my experience, at least with soups and sauces.