Learn Croatian: Nisam izgubio tvoj novčanik.

Because novčanik (wallet) is a masculine inanimate noun (a non-living thing). In Croatian, masculine nouns that are things don't change their endings in the accusative case; they look exactly the same as in the nominative (subject) case. The word tvoj (your) follows the same rule.

How would I say "I didn't lose it" using a short pronoun instead of "your wallet"?

You would use ga (him/it). Because novčanik is a masculine noun, you must use the masculine accusative clitic pronoun. The sentence would be Nisam ga izgubio. Notice how ga comes right after Nisam.

Does the word izgubio tell us if a man or a woman is speaking?

Yes! The -o ending on izgubio tells us the speaker is male. If a woman were saying this, she would use the feminine past tense form and say Nisam izgubila.

Why don't we need the word ja (I) at the beginning?

Croatian often drops subject pronouns like ja because the verb already makes it clear who is speaking. The word nisam specifically means "I am not", which is used here to build the negative past tense for "I".