No. Translate for sentence "I have my own dog." is "Mám svého vlastního psa."
Both "Mám svého psa" and "Mám svého vlastního psa" mean "I have my (own) dog"; in this case "vlastní" only emphasizes that the dog is really your own. However, the pronoun "svůj" by itself already means "one's own" (more specifically, it refers to possession by the subject of the sentence), and it can be used to clear up some ambiguities (as a side note, "svůj" is analogous to "sin" in Swedish). For example:
Irena ztratila její knihu = Irena lost her (another person's) book
Irena ztratila svoji knihu. = Irena lost her own book
Notice that, in English, the sentence "Irena lost her book" is ambiguous (we don't know whose book Irena lost), but the two sentences above (in Czech) aren't.
The word "pes" is nominative; the word "psa" is accusative, which you need here because it is the direct object of the verb.
So I wrote "I have their dog", which is noted to be wrong. But if I tap svého in the sentence, it shows "my, theirs, his" as a translation?
The word "svého" always refers back to the subject of the sentence. For example, if the subject is "he", then "svého" will mean "his". But here the subject is "I', so here "svého" means "my".
Another way to look at it is to think of "svého" as meaning "own": He has (his) own dog, I have (my) own dog, etc.