Before I go on to address your specific questions, I should state that formalin is not my favorite treatment option. Using hyposalinity (given an accurate glass, floating-style hydrometer or refractometer) or even daily water changes are my general recommendations.
"Is it possible to carry out this treatment using the hospital tank water and then discard after a bath." - Sure. That would be my preference. Isolate the infected fish in a QT tank, draw water from that tank every other day for a formalin dip, and then replace with new, well mixed saltwater.
"After two weeks of treatment all of the organisms should be dead on the fish and in the hospital tank-correct?" - Correct
"If so it should be safe to net the fish from the last dip and place in the display tank-correct? Or should we wait for another two weeks of observation in the hospital tank?" - I would prefer to hold the fish in QT after the treatment is complete for an additional month to ensure optimum health.