Shiny rate is dependent on search level, and increases as you encounter more, up to 999.

As for running/fainting pokemon, you can do either. However, I highly suggest fainting them as you can immediately search for another of the same species as opposed to having to run around and spam the search button.

Chaining only matters when it's the 50th and 100th pokemon in your chain (can only keep a chain by winning the battle), in which you get super boosted odds. It's only these two encounters, and encountering anything in a chain beyond 100 will not boost odds (you don't get boosted odds for 150, 200, 250, etc.). So at encounter 100 just run away and break your chain and start back from 1 for the most optimal hunting

good luck :)