I can't speak about regrowing these in water, but I will share what I do which is similar.
I regrow these by cutting them off about half an inch above where the white section ends and planting them directly into the ground. This might solve the issue you mention in point 1.
Once planted in the ground you can continue to harvest by cutting back and take what you need leaving the plant in the ground to grow. However, one problem with doing this is that the new growth will not grow back to the same size as the original plant so you gradually get less and less from the plant. I would suggest you can only harvest another 3 or 4 times like this until the regrowth is too small and thin.
Another problem with using the root for replanting is that the root has the strongest flavour. So from the first use you are already discarding the best part.
Considering that a bunch of these costs about $1 you are not doing much in terms of sustainability. I would suggest if you wanted to save money then I would go out and buy a packet of seeds.
However, if you can be bothered to put in the effort, it is actually worthwhile if you can get 100 of these planted in the ground together. It uses very limited space and it is very convenient to be able to take these as needed rather than going to the store if you run out.