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the electric field - when there's no electric charge - is by definition 0. But in nature, there are charged particles and changing magnetic fields everywhere so yes - there's usually always some kind of electric field...
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yes. electrons, quarks... have a electric charge, thus creating an electric field.
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Yes, in plasma which is 99.99% of whole universe for example: sun and interglactic space. they electrons and ion(+ve charge) combine up to give total charge neutrality "in bulk", but coulomb force between them does not exist, which is condition of existance of plasma. In one line " our unierse is electrical". even if charge in bulk "may" be zero, the energy transfers in form of electric current.
Yes electric field really exist. It's a region where electrical effects can be experienced like any charge particle in that region experience a force. This region is around any charge particle.