Water – Any idea what parts were replaced on the water softener

1) The crystals are most likely deposits left from the evaporation of water that used to be in the tank (as you mentioned, you had the lid off). As the water evaporated, the dissolved minerals transitioned into their natural crystalline structure, which is attractive to the dissolved minerals -- rinse and repeat and you have a larger crystal structure adhered to the wall.

2) You could -- but from the look, my personal preference would be to empty it and give it a good and thorough cleaning.

3) A thorough cleaning once a year is often recommended. Is it necessary? Maybe not, but it couldn't hurt anything either. I'm sure your owner's manual outlines a deep cleaning procedure, but if not, there are a number of resources on the internet.

Regarding the disinfection -- I would follow the manufacturers recommendations. You should use a fragrence-free bleach. Store-brand bleaches often have less additives than Clorox, which is good when the use is "food"-related. Clorox uses additives which aid in the removal of soils (things like caustic soda, soda ash), surfactants, and other ingredients to help prevent stains (sodium polyacrylate). 4 oz does seem a bit much, but the manufacturer would know better than me what should be used.

4) I'm not sure why your manual does not recommend the mixture of potassium chloride and sodium chloride -- these are mixed in water softeners quite often to decrease the saltiness of the drinking water and save a bit of money. My vote is, when in doubt, follow the manufacturers recommendations -- they only spent millions developing the thing (of course, they would also like to make money when you need to replace it, so it's a bit of a double-edged sword).

I only see one valve, incorrectly installed in the wall cavity at that, so there is no way to bypass the softener for maintenance or if a component fails. So no, its not installed correctly because a bypass is a standard part of any softener installation.

Without a bypass you are going to find that those cheap plastic components they make water softeners with are going to eventually fail and it will spring a leak. Hopefully the incorrectly installed valve you have now comes before the water reaches the unit. Then you'll at least be able to shut the water off from right there and not at the curb. However you'll then be without water in the house until you fix whatever issue your having with the softener. Below is a diagram of a standard bypass setup which your manual should also have.