I think "Mám psa." is correct too.
I have a dog.
Why is svoje not correct
You need to use the correct grammatical case - accusative here.
Is there any place online where this is a bit more documented? Cant find it on duolingo
Read the Tips provided. Use any web browser (even mobile) and after selecting a skill click on TIPS.
Other than that, there are many other external online resources about Czech grammar.