"Haustiere mögen nicht Kinder"-is not correct. What is the rule of placing this "nicht"? Sometimes it appears after verb sometimes not...
What is Keine?
It is the topic of future lessons and it is not so simple to explain it here because it also means "no" as "nein". You will learn it completely at the end of second cycle. For now look to the above link in the comment of Hohenems.