it must be hard for European women to live there without unintentionally causing offence, maybe breaking the law.
Not at all. It's not Saudi, and I think some responses on here seem to be conflating Dubai and Jeddah. It's perfectly possible to live a largely western life there, whether male or female, keeping your head down, and not drawing the attention of the authorities. (I did, for a couple of years.)
Dubai, of all the emirates, has put a lot of effort into looking western-facing and liberal, to the point where I doubt many of the western holidaymakers who go there know it's essentially a tribal dictatorship.
Even Emirati citizens (a tiny minority of the population, which is mostly expat) don't have a vote, and there are no political parties -- apart from the seven hereditary emirate leaders, and the family members they appoint into key roles, there is only since 2006 a 40-member Federal Council, half of which is appointed by the government, and whose role is only advisory. It's a country run by a series of wealthy, powerful, intermarried, very patriarchal families.
There are restrictions on freedom of association and assembly -- NGOs need government permits to operate, textbooks are censored, public meetings need official permission, there is no right to strike against poor work conditions. The justice system (divided between Sharia and civil law) is not independent. All legal decisions are subject to government supervision. Emirati dissidents asking for greater power for the Federal Council have been imprisoned or had their passports confiscated, and the UAE helped suppress Bahrain's independence movement during the Arab Spring.
There is no freedom of the press -- media outlets may not circulate “defamatory material and negative material about presidents, friendly countries, [and] religious issues” or face huge fines. And of course journalists can be deported at no notice anyway.
So Sheikh Mo doesn't need 'good PR'. No one inside the UAE will publish anything negative about him and his family.
Sheikh Issa, the half-brother of the current ruler of Abu Dhabi, was filmed torturing and sexually assaulting an Afghan man the footage is still publicly available and was found not guilty by an Abu Dhabi court. This caused curiously little international outrage.
The fact is that the UAE is considered western-friendly and a key anti-terrorist ally in the region, so there is very little political will to cause a fuss on 'internal matters' like the disappearance of a royal family member.