Learn Polish: Proszę, to jest kawa. - Here you are, this is coffee.

You are right! Proszę is the standard word for "please", but it is also the exact word you use when physically handing something to someone. In English, you would say "Here you are" or "Here you go", but in Polish, you simply say proszę.

Why do we use to (this) instead of ta? Isn't kawa a feminine word?

When you are introducing or pointing out what something is using the phrase "this is", you always use the fixed phrase to jest, regardless of the gender of the noun that follows. You only use the feminine ta if you are describing a specific object's trait, like saying "this coffee is hot" (ta kawa jest gorąca).

Why does kawa stay in its basic dictionary form here? Don't we usually change word endings after the verb "to be"?

It is because of the word to. When you use the specific structure to jest (this is), the noun that follows stays in its basic dictionary form. If you used a regular pronoun instead, such as saying "she is a woman" (ona jest kobietą), you would have to change the noun's ending.

Can I drop the word jest and just say to kawa?