1) I reckon it's likely that you are generally on the right
track. This is something a lot of textbooks get wrong.
2) It would help to spell out in plain English the background
of what you are doing, and the objective.
3a) If I had to guess, I'd say there's an experiment to measure
the half life by counting decays, and it's tricky because:
— The number of decays in any reasonable interval is small.
— The observed numbers are subject large-percentage fluctuations
due to Poisson statistics, even though the underlying physics
is not changing.
— Ordinary "textbook style" least-squares curve fitting to
extract the rate fails miserably. That's because "least
squares" usually means maximum likelihood, which is jargon
for maximum a priori, but any sane person would want maximum
a posterori.