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Kekkei Genkai translates to blood inheritance limit, but is better described as a genetic trait. An even better description would be a small segmentof the genetic code that creates a unique chakra and sometimes a bodily effect.
Chakra is created in the organs, and is fed throughout the body through a chakra network. A persons genes dictate how these organs develope and function. A chakra based kekkei genkai (like mokuton and hyouton) is created when a persons organ produce two chakras in near perfect equality. This is why these kekkei genkai are so rare, because people rarely have a natural affinity to more than one chakra, as their genes tell their organs to produce more of one chakra than the rest. This is why Naruto has a wind natured affinity, and the Uchiha have a natural fire affinity. Nagato, with his Rin'negan eyes, has a perfect balance of all chakra types, including yin and yank. Mixing the two chakra's may require a catalyst as seems to be in Haku's case. Yamato said that he needed use water in one hand, and earth in the other. But Haku was able to use one handed hand signs (Which would probably be hard enough with one chakra type) to create hyouton, which means that he mixed wind and water chakra BEFORE he molded it. This pre-mixing may have been aided by a hormone or catalyst.
The Shodaime Hokage's Mokuton may not have been passed down do to an inability to reproduce (that would suck), or by his choice not to have children. Tsunade is his great grandniece, which means she isn't a direct decendant.
Kekkei genkai that are visibly present (doujutsus, and Kimimaro's ability) are created by a gene that creates tissue variations.
The Sharingan is only visible when the user focuses chakra into his eyes, causing the eye (Iris? I don't know my eye anatomy very well) to change color and functionality, by heightening the strength of the cones and rods in the eyes. The ability to percieve chakra flow may be aided by an extra unnamed receptor unique to the sharingan eye. Kakashi's eye is permanently in sharingan form, which means that his eye is always using chakra, but because he doesn't have a proper balance of chakra type (his genes don't produce the right amount of whatever chakra type needed. Likely fire) needed for the eye to function efficiently, he ends up using more chakra than an Uchiha would. The sharingan also changes the number of tomoe, depending on how much of each chakra type is being used, or even how much of it is used. An extreme emotional response, may cause hormones to produce large amounts of the needed chakra, causing the persons eye to either take on the form of sharingan (any number of tomoe in either eye. The body may not symetrically use the chakra resulting in eyes with different numbers of tomoe), increase the number of tomoe, or even take on the mangekyou form. Exceessive use of the mangekyou sharingan, by the original owner of the eye, may wear out the cones, rods, and chakra receptors, causing blindness. When an eye transplant occurs, the eyes are straining to obtain the desired amount of chakra, instead of over using it, creating an "eternal" MS. This may mean that Kakashi may only be weakened by the lack of chakra, instead of going blind like Sasuke and Itachi. Kakashi may have awakened his MS by using a hand sign that focuses the right balance of chakra into his eye.
Kimimaro's ability occurs when his body rapidly uses a large amount of chakra that results in increased bone developement. The cursed seal causes his body to produce even more chakra, resulting in his tail thing, and the spikes on his back during the second stage.
The Byakugan eye is in constant effect. The chakra receptors in the eyes are located all around the inside of the eye, except for the eyris, creating the blind spot, and the ability to see in near 360 degrees. This would be limited to seeing chakra only, where the user only sees the visible spectrum through frontal vision. When the byakugan is used in extreme, the users veins bulge around the eyes, causing large amounts of chakra to flow all around the eyes, and enhancing the users perception of chakra allowing the user to see the chakra points in the body.
The Rin'negan eye is unique, even to the world of kekkei genkais. It allows the user to create a perfect balance of all chakra types, and to likely percieve them too. The user probably sees the different chakras in different colors. But more importantly is the production of all chakra types in balance. Each chakra probably resonates at a certain frequency.
Peins bodies are covered with chakra sensitive piercings. But these piercings are in specific locations on both Nagato's body and the Pain bodies. This may be because Nagato is sending and recieving chakra signals from certain body parts, allowing him to see through ALL of his bodies, as well as make them move. This is why all of his bodies are dead, so the bodies don't interfere with the chakra signals. This is also the likely reason why only the god realm talks; a tongue piercing. You may now be wondering how Nagato controls each body seperately, and only up to six bodies at a time. Well the answer is chakra frequency. Each body is sensitive to only one chakra type, meaning the signals sent by Nagato are each in different frequencies, with each frequency (one for each chakra type, inclunding yin/yang), are only effective in the bodies most sensitive to them. If Nagato used two bodies with the same chakra affinity were used by Pain, both bodies would move and react the same, plus the sensory signals would merge creating inaccurate sensations. Nagato is able to see with all six bodies because each body sends back visual signals of different frequencies. His body would then percieve different sights by focusing on one frequency. This would require a lot of brain power at first, but given time and practice, Nagato was able to get the hang of using multiple bodies at once. In theory, Nagato would likely be capable of using Hyouton, Mokuton, and lava (youton?) techniques, but not through the 6 paths.
In theory, anyone can use a kekkei genkai, the only thing stopping them is their chakra affinity. It would take a large period of time and luck to use any kekkei genkai by a person who doesn't posses the genes that assist in creation and use of kekkei genkais and the chakra's needed for them.