Are all entities of mythologies believed in, and if so, for what reason?
I wouldn't say all. Most people nowadays focus on the 'big' figures such as the Gods although other 'lesser' figures such as Pegasus and heroes such as Heracles have their own story to tell. Belief is a tricky word. Magicians have always emphasized the empirical results obtained through magical and mystical practices without reference to particular beliefs so I don't think belief is relevant.
If they are, can different names describe the same entity? (Zeus/Jupiter and the like)
Most often entities have a variety of names. Jupiter had dozens of epithets and so did a variety of other gods. In the Judaic-Angelic tradition one of the chief angels, Metatron, has close to one hundred or so names (if I remember correctly).
The general theory regarding entities such as Zeus, Jupiter and other similar sky father-gods is that they are particular cultural expressions of universal Ideas or Archetypes. These Ideas or Archetypes in themselves are not considered as entities in themselves but rather universal reservoirs each having a vast array of patterns. Regarding your question I'd say different names can describe the same Idea or Archetype as opposed to different names describing the same entity particularly when considering deities. While Jupiter, Zeus, Amun, Indra, etc. might share similar attributes it'd be a mistake to think their names were synonymous with each other. They have nuances that set them apart from other deities presiding over the same phenomena. The inverse is also true. There can be more than one entity with the same name as another.
Is it possible to have a relation with a very specific natural element? (i.e. the East Wind)
Yes. In fact it's a requirement to work some kinds of magick.