There are Five Elements: Earth, Water, Air, and Fire, plus Ether. Ether is force. Where the Four relate to states of matter, namely solid, liquid, gas, and plasma. The plasma is mainly the heavenly fire of the sun, and aspects of lighting, so Fire includes Radiant and Lightning. The Ether relates the 'fundamental forces' that are immaterial but physical, including gravity. However, the Fire-Radiant-Lighting also covers any of the electromagnetic force. Ethereal force includes psionic telekinesis, whence flight and objects made out of force.
It often works out well, to treat Plant as a living Element. Plant is sometimes synonymous with Air, in the sense of Daoist expanding-encompassing motion, in the sense of Norse Yggdrasill heavens, and sometimes synonymous with Earth, in the sense of springing from the ground. Plant can also have watery and fiery (sunlight, burning) aspects. For D&D, it makes sense for Plant to be a Living Element composed of the other Four Elements and associating the Material Plane where the other Elements recombine.
The easiest way to handle the Elemental themes is to reorganize the entire spell list according to them. Then any class can be assigned an element, by treating it like a spell school or spell domain.
For utility, including to ensure there are decent spells for every slot level, an Elemental character concept should choose at least two from the Five Elements.
Even if we aren't going with the 4e cosmology, where the gods and elemental powers are explicitly at odds, I feel like the delineation still exists. Divine domains are conceptual and based around the lives of their worshipers, while the elements are more of an abstract building block..
Fire is not a concept. You might see fire used by a God of the Sun, or a Goddess of Home and Hearth, or a God of the Forge. But you don't get a God of Fire any more than you get a God of Roads. Roads fall under the God of Travel, which is a nicely broad concept, while roads are a specific thing.
Yes, and it's important to bear in mind that different cultures have different conceptions of what constitutes an "element" or "elemental force" as well.
The four elements of hermetic magic and classical Greek philosophy are the ones we're most familiar with from Greece's prominence in Western Civ, but they're not the only take on the elements.
Japanese philosophy, for example, referencing Earth, Water, Fire, Wind, and Void.
Or Chinese/Daoist/
Wuxing having wood, fire, earth, metal, and water.
Or the pre-Christian Irish having three realms making up the world (Land, Sea, and Sky; fire is of the gods), and nine elements or Duile which make up the world and have correspondences in the human being- Earth (flesh), Stone (bones), Plants (hair), Wind (breath), Water (blood), Clouds (mind), Stars (senses), Sun (face and outward aspect), Moon (the soul).
You can see an interesting correspondence to the Irish elements in Norse myth, where Odin, Vili and Ve made Midgard (our world) from various parts of Ymir's body- Ymir's flesh for earth, bones for mountains, blood for rivers and seas, hair for trees and vegetation, brain for clouds, skull making the sky, etc.