Desk research against primary sources, with the issuing authority linked where a page states an official fee, rule or deadline.
Legality since 2019 covers the product itself — TRA approval — separately from where you are allowed to use it, and the second question is answered by simply mapping vaping onto the existing smoking rules rather than by any vape-specific law. If a space bans cigarettes, it bans vaping too, including enclosed public spaces, restaurants, government buildings and anywhere near schools.
The customs limits are the part people miss because they assume legal in Dubai means no import limits: the 60ml e-liquid cap and the 2-device maximum apply on arrival, and going over either without declaring is a customs matter distinct from the AED 500–3,000 fine range for vaping somewhere it is banned. Declare before you are asked rather than after — the two issues get resolved very differently depending on which side of that line you are on.