Does Running from a Pokemon Break the Chain?

Encountering another Pokemon while in a chain will not disrupt the chain unless you catch or faint the Pokemon that is not the one you are chaining. Running away from an encounter whether it's the Pokemon you are Chaining or not will not disrupt the Chain but will not increase it either.

Does running away break a catch combo?

Your combo will end if you catch a different species of Pokémon, or if any wild Pokémon runs away from you. Your streak will not end if you flee the encounter instead, so you should bail if you accidentally bump into the wrong Pokémon.

Does running away reset shiny chain?

If you stumble into one that isn't the same species, make sure you run away and don't catch it or let it faint, because your chain will be reset. Once you hit a chain of 25, that's the maximum you can shiny chain in Sword and Shield.