Learn Croatian: Imam sina.

In this sentence, the son is the direct object (the person being "had"), which means the word needs to be in the accusative case. For masculine nouns that represent living beings, like people and animals, the accusative case adds an -a to the end. So, the base word sin becomes sina.

Would a non-living masculine word also get this -a ending?

No, and that's exactly why we call this the animate masculine rule! If you were talking about an inanimate (non-living) masculine object, the accusative case wouldn't change the word at all. For example, "I have a watch" is Imam sat, not Imam sata. The -a ending here is strictly for living masculine nouns.

Where is the word for "I" in this sentence?

The verb imam already means "I have". In Croatian, the verb endings tell you who is doing the action, so you can usually drop pronouns like ja (I) unless you want to add special emphasis.

How do we know it means "a son" and not "the son"?

Croatian doesn't use articles like "a", "an", or "the". The word sina can mean "a son" or "the son" depending entirely on the context of the conversation. Here, "I have a son" is just the most natural translation in English.