Learn Italian: Ecco il caffè, signora. - Here is the coffee, madam.

The word ecco is a special pointing word that means "here is" or "here are" all by itself. You do not need to add a verb like è (is) or sono (are) after it.

Could I use qui instead of ecco to say "here"?

Not in this case. Qui simply describes a physical location (as in "The coffee is here on the table"). Ecco is used when you are presenting, handing over, or pointing out something to someone, just like saying "Here you go!" in English.

Why does caffè have an accent mark at the end?

In Italian, an accent mark on the final vowel tells you to put the stress of your voice on that last syllable (caf-FE). Without it, you would normally stress the second-to-last syllable by default.

Shouldn't it be la signora instead of just signora?

Because we are talking directly to the woman, we drop the word la (the). You only use la signora (the lady / Mrs.) when you are talking about her to someone else, not when addressing her directly.