Hardwoods with a high tannin content produce a black stain if in contact with fixings that are starting to corrode. Tannin doesn't cause fasteners to corrode, the tannin reacts with metal that is corroding through exposure to water. The black stain is called "iron stain" and is explained in a data sheet that can be downloaded via this link https://www.fs.usda.gov/research/treesearch/48449.

We agree, it's very difficult to maintain a complete seal. Even if the bearers are pre-coated all round, the coating may be damaged during installation. Also movement in the timber could break the seal and create an opening for water to enter. It's best to leave the timber uncoated so it can dry out as quickly as possible when the water level drops. Note that all fasteners will need to be stainless steel. You don't mention the type of timber. Preservative treatments have limited penetration into hardwoods and can only be relied on to protect the sapwood.

If the stringers are fixed to wall studs both sides, essentially the stringer span becomes the distance between studs, measured on the angle. So the short answer is yes, the stringer size could be reduced, consistent with having enough width to accommodate the treads. However you would also need to be sure that the stud walls can accommodate the extra load imposed by the stairs, ie. the walls will be taking more load than they would be if the stringers were spanning the full 4.3m without being fixed to the studs.