Question about anata

I recently found a really great penpal friend from this site about a week ago, her name is Chisato. I was wondering, when I want to ask questions like "do you have any pets?" should I use her name instead of anata? Should I add a suffix to her name?
She seems to use friendly/informal Japanese in her emails to me, so I don't want to come off as too formal.

I'm sure you know but we Japanese don't usually use the word "anata" in questions like you do in English or other language. If you use her name instead of "anata", it will sound as if you are asking about a third person so I don't recommend it. Hope this helps you!

If the subject is clear Japanese omit it. Since you guys talking about yourselves, it's obvious you are asking about her pet. "ペットを飼ってますか？" is fine.