Learn Portuguese (Portugal): A mulher vive, em média, mais do que o homem.

In Portuguese, using the definite article with a singular noun is a common way to make a general statement about an entire class or species. While you could say As mulheres vivem..., using the singular A mulher gives the sentence a slightly more formal or universal tone.

Why use mais to mean "longer"? Could I use mais longo here?

You cannot use mais longo here. Longo is used for physical length or the duration of specific events, like a meeting or a movie. When talking about doing an action for a greater amount of time, such as living, sleeping, or waiting, Portuguese simply uses mais (literally "lives more").

Is the do in mais do que required, or could I just say mais que?

You can drop it. In Portuguese comparisons, do que and que are generally interchangeable when they come before a noun or pronoun. Both mais do que o homem and mais que o homem are perfectly correct, though keeping the do is very natural and common in spoken European Portuguese.