Learn Polish: Czy widzisz tego kota?

Ten kot is the basic dictionary form. But here, the cat is the direct object (the thing being seen), so it needs the Accusative case. Because a cat is a masculine living thing (animate), we add special endings: ten changes to tego, and kot adds an -a to become kota.

If I were looking at an inanimate object instead of a cat, would the endings change?

No, they would stay the same as the dictionary form! For masculine inanimate objects (non-living things like a table or a phone), the Accusative case looks exactly like the basic form. So 'Do you see this table?' is Czy widzisz ten stół?. We only use tego [word]-a for living masculine nouns.

Does czy translate directly to 'do' in English?

Not quite! Polish doesn't use helper words like 'do' or 'does' to make questions. Instead, czy is simply a question word placed at the start to turn a statement into a yes/no question. You could actually just say Widzisz tego kota? with a rising tone of voice, and it would mean the exact same thing.

Why isn't the word ty (you) in this sentence?

In Polish, the verb ending usually tells you who is doing the action. The -isz ending on widzisz means 'you see'. Because the verb makes it clear, the pronoun ty (you) is usually dropped unless you want to put special emphasis on who is looking.