Thanks in advance.

1. Is Dubai under Shariah (Islamic) Law? or a different law exists for this emirate?
2. Is it true that sharing of a flat/villa/apartment between a male and female is not allowed?
3. Is it true that a bachelor cannot rent his own flat/villa?
4. I've seen articles that public display of affection is considered "haram" in Dubai.
If a male and female go out on public together, what "acts" are not allowed?