My deck needs to be restained, but before I do that there are some
rotten boards that have to be replaced. These boards are rotted in a
corner at the end where they meet a step up. That corner is shielded
from sunlight and wind so leaves gather there and stay wet, hence the
rotting.

My deck floor is made out of 2x4s rather than deck boards. I'd prefer
to just cut back the rotten boards to the nearest joist(?) and put in a
new piece. If I go that route how far back should I go to ensure
stability? The nearest joist seems to be a foot away. Would it be
better to go back two full joists?

I assume treated lumber is necessary so the question arises whether
this will accept stain. How do I prepare it for stain and hopefully
make it match the rest of floor?