Unfortunately not. A will must be registered personally, because it requires attestation and authentication of your signature.
No, that would not be possible. In case a will was not registered before death, the assets will be distributed according to Shariah and expatriates will need to go to a court in order to enforce foreign rules on wealth distribution.
No, it is not mandatory, but will registration in Dubai is advisable. It will help you to avoid future expenses on lawyers and waste of time in the courts.