Yes, the white lines around the tile grout is calcium leaching out. Really can't stop that from happening.
Dilute acid 50/50 with water in a plastic squeeze bottom and squirt onto the tile grout to perhaps remove the brown. Be careful to not add too much acid at one time. And check the pH afterwards, and adjust if needed.
The spot in the plaster might be from within the plaster and might not be able to be removed, but you could also try the acid also on that spot.