For example, if I encounter a Charmander, see that it is not shiny and exit the encounter, will the odds of a shiny Charmander appearance change on my next encounter?
No. The chance of a Pokemon being shiny to an individual player is specific for each Pokemon, and doesn't change based on what Pokemon you've encountered before. This rate is observed to be about 1 in 450.
In fact, it's very common for people to enter the catch screen to see if it's shiny, then flee when they realise it isn't, and repeat for all the Pokemon in the area.
The factors that seem to increase the chance of a specific Pokemon being shiny or not are:
- Legendary raids where their shiny Pokemon are highlighted e.g. Lugia, Ho-Oh, Kyogre, Groudon
- Raid-exclusive species e.g. Mawile and Absol
- Specific events such as Aerodactyl raid bosses during Adventure Week
- Community Days
- Baby Pokemon
- Field Research
Note that many events that introduce Shiny Pokemon do not increase the shiny rate.