Pokemon doesn't have many canon explanations for these, as a reuslt, some of them are often left to the fandom to decide. A lot of these are also assumptions and guesses on my part to apply logic to the world.
First paragraph: It's a show aimed at kids mostly, same as the games. Characters don't look 10, but rather cna fit into most early age brackets. And yes, Pokemon is anime.
There is no logical explanation behind Pokeballs, other than the diffusion of matter. In my stories, the Pokeball contains technology capable of transferring physical matter into a pure energy form, thus allowing the Pokemon to be stored within the Pokeball. As all amtter is simply different forms of energy, the conversion is easy enough to say in theory, but hard ot put into practice. Additionally, Pokeballs can expand or reduce in size due to the technology of the Pokeball moving around, thus allowing it to enter a more compact mode. It's unsure what the inside of a Pokeball is like, seeing as the being inside is theoretically pure energy, but it is often guessed, and supported by Nintendo, that the inside of the Pokeball varies to give the mind of the Pokemon, or the physical body if the energy process turns it to energy, then to matter again inside the Pokeball, a more sane level of existance. A water type may need water, thus the Pokeball compensates thruogh inbuilt technology. In this thread, of the inside of a Pokeball is chosen to suit the Pokemon, it also follows that the environment is suitable for the Pokemon to rest and heal. Given the choice, I suppose a Pokemon may even be able to listen to events occuring outside its Pokeball, giving the possibility that the Pokeball has psychic capabilities and responds to the thoughts of a Pokemon.
Within the anime and games, Pokemon do need additional medicine for particularly extreme wounds or conditions (Poisoned, as an example), but other than that, Pokemon have probably developed accelerated healing abilities as a result of their frequent practice of battling. Pokemon have aso been shown to be fed regularly outside of the Pokeball, showing that the Pokeball doesn't seem to provide nutrition. As a result, they can starve to death unless they are Ghost Types.
Pokemon do not evolve constantly, as each tends to have one, or at most, two forms to evolve into. Humans are constantly evolving in their childhood to their teenage, and then adult forms, it's just more gradual. In the case of Pokemon, these changes are more sudden. It's the training that allows a Pokemon to evolve, similar to how exercise and nutrition allows the human body to grow towards adulthood. It's quite similar to the human developement, just occurs at set stages that can spontaneously happen, whether upon an emotion trigger or the need to evolve to deal with their current situation (See when Ash's Charmeleon evolved to Charizard to save Ash from the Aerodactyl). The elemental stones are shown to contain concentrated doses of whatever element they represent, which leads to the suggestion of radioactivity, a natural element not found on earth or even divine blessing.
Eevee is capable of evolving because it is very adaptable. It has been specifically noted that Eevee's DNA is unstable, allowing it to potentially absorb part of the environment around it and evolve to best suit it. This would explain why Eevee evolves around the Ice Rock in Diamond and Pearl, or to Vaporean when exposed to a Water Stone.
Pokemon attacks can be a bit tricky, but again, I also dealt with this in my own Pokemon fanfic. The water used by certain water Pokemon (Squirtle, Totodile, Mudkip, etc) is developed in an internal water pouch which frequently fills with either excess saliva, stomach acid (Explaining why several Water Types are also poison types), or is capable of combining oxygen in the air with hydrogen produced naturally by the body.
Leaves and vines are harder to explain. However, there are cases where plants can be quite effective against regular metal. Theoretically, by binding together thousands of paper thin layers of membrane, you can create a substance equal in power to steal. This would be seal in a laquer naturally product by grass Pokemon. The vines may simply be the result of seeds carried by the Pokemon being forced to grow rapidly, or even the Grass Pokemon itself forcing its body to grow by concentrating sunlight, nutrients and water in its body to force the growth or attack (Solarbeam requires sunlight to fire, etc).
And that's all stuff I've dealt with in my story. It's actually quite fun, applying logic, you just gotta ask 'How can this work', and then create the theories. I dealt with the paradox of 'Pokemon fire = Not Death' vs 'Building on Fire = Death' by saying that fire types excrete a natural chemical in their fire sacs which limits the heat their flames can reach.