I answered with "He gives her tip". The correct answer is given as "He gives her a tip". Would "He gives her the tip" also be correct? How do I know if "a" needs to be included, or if "the" can be included, in this situation? Why is the German not "Er gibt ihr einem Trinkgeld" or "Er gibt ihr das Trinkgeld"?
Would "He gives her the tip" also be correct?
Not as a translation of Er gibt ihr Trinkgeld; if you are referring to a specific quantity of money that the listener knows about ("the tip" rather than simply "a tip"), it would be Er gibt ihr das Trinkgeld in German with the definite article.
How do I know if "a" needs to be included
Basically, you have to know whether a noun is countable or not.
"tip" is considered countable in English, but Geld is uncountable in German and Trinkgeld almost always is as well -- thus we usually say Trinkgeld rather than ein Trinkgeld for "a tip".
Usually, countable nouns in English correspond to countable nouns in German, but not always; "tip" and Trinkgeld is one counterexample, and another famous one is Gemüse (uncountable in German, but plural -- and hence countable -- in English: "vegetables"). Möbel / "furniture" is the other way around: plural in German, uncountable (and singular) in English.
Just things you have to remember.
Why is wrong: He gives her tip
Because it has to be "a tip" in English.
Trinkgeld is a mass noun in German: uncountable and grammatically singular (like "money").
But in English, "tip" is considered countable, so you have to say "a tip" rather than just "tip" or "some tip".
Is trinkgeld = tip?
Yes and no.
First of all, there is no German word trinkgeld. It's Trinkgeld with a capital T, since it's a noun.
Second, Trinkgeld is (usually) uncountable, while "tip" in English is countable.
So you can say "She received a lot of tips that weekend", but you wouldn't say Sie erhielt an dem Wochenende *viele Trinkgelder; it would be viel Trinkgeld instead. Like how you would say "a lot of money" (singular, uncountable) in English and not "a lot of moneys" (plural).
So, Trinkgeld (capitalised) is often "a tip" (with article), sometimes "tips" (plural, no article).
The two words are not exactly equivalent.