Shandini, with respect, but only in Pprune do you get non pilots jumping in with answers to flying questions of which they have no concept. You would have done better to keep out and let pilots answer! It's like an accountant watching a Doctor forum and stepping in with advice on obscure procedures! At least you gave a forewarning!
An active fire warning: even ceased, would have me landing as soon as poss.
Flame out: discuss with engineering possibly, but land.
Consider overweight go-around performance on one engine- not funny.
Sick passenger: depends on seriousness. If you start weighing the safety of 300 people against the health of 1, you are possibly already making a misjudgement.
No special guidance. Left to the experience and judgement of the pilot