Learn German: Ich finde hier kein Handtuch. - I can't find a towel here.

The word Handtuch is the direct object of the sentence (the thing you are finding), which means it is in the accusative case. Because Handtuch is a neuter noun (das Handtuch), its negative article in the accusative case is kein. If it were a masculine noun, you would have to use keinen.

The translation says "can't find a". Why doesn't the German sentence use nicht for "not"?

In German, when you negate a noun that would normally use "a" or "an" (like "a towel"), you must use a form of kein (meaning "no" or "not a"). You cannot say nicht ein Handtuch. Literally, the German sentence translates to: "I find here no towel."

Why is hier placed before kein Handtuch?

In German word order, adverbs describing place (like hier) or time usually come before the direct object if that object is introduced by an indefinite article (ein) or a negative article (kein).

Is Handtuch a compound noun? How do I know its gender?

Yes! It is made of die Hand (the hand) and das Tuch (the cloth). In German compound nouns, the very last word always determines the gender. Since it is das Tuch (neuter), the combined word is also neuter: das Handtuch.