So the noun "dog" is treated the same as a masculine person in Czech noun cases?
Yep. We treat animals the same as humans, the important thing is that they are animate. Note: Trees are inanimate, something I disapproved of when I was a kid. :D
How about tomatoes? For some reason tomato, pomidor is treated as an animate noun in Polish hahaha
Vidite is verb "to see". No "can" verb around - like umět or musět - still not sure which one applies. Why the answer "Have you seen my dog" is not valid?
"Have you seen" implies the past. Vidíte is present tense. It should be "Do you see my dog?"
In every day use what is more common or better to say? Vidite meho psa or vidite moje pes?
The form "moje" would be correct with a neuter noun in the accusative singular, but "pes" is masculine, so must use "mého" in the accusative singular.
It's definitely the first option. The second one is not grammatically correct. The latter would work in plural (if you changed it a bit). If you had more dogs, you could ask: "Vidíte moje psy?"