Can I say widzę pies?
No. Direct object requires (usually) the accusative case - 'widzę psa' is an example of this. In negative sentences and after some verbs you use the genitive case, but you never leave it in nominative.
Why so many names for a dog( pies) in polish? What becomes of "Mężczyzna"?
It's just one 'name' in different grammatical cases. For "I see a man", it would be "Widzę mężczyznę".