"I have a computer."
In Hungarian the word order is generally free. This means that especially for basic sentences, every, or almost every possible order is correct. The thing is, that they all differ in their meanings, usually in the emphasised word. Therefore I can't say that any of the sentences you've met are wrong, but they might be rare sentences only used in special situations.
You do not necessarily need a second sentence. You could be sitting in a classroom where the teacher asks everybody to say what they have.
Nekem kutyám van. - I have a dog.
Nekem biciklim van. - I have a bicycle.
Nekem számítógépem van. - I have a computer.
All these are emphasizing the thing/object. Because that is what was placed in front of the verb.