"Это самая трудная задача."
Translation:This is the hardest problem.
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Проблема is a generic problem, like a meteor falling or your car breaking while you're late to work. While задача is a task you have to fullfill. It doesn't necessarily have to be a problem.
I guess completing your daily dose of duolingo Russian course can be a задача (you're completing a task), while not doing it is a проблема (you've learned nothing, the big owl is watching you).
Not always. It's often superlative. It can also be used in a couple other ways. Wiktionary is a good resource as well. It also explains the definitive pronoun use. https://en.m.wiktionary.org/wiki/%D1%81%D0%B0%D0%BC%D1%8B%D0%B9