The chromate conversion coating for copper preserves the look as far as I can see. In general you would want to look at "bright dip" coatings I guess. Commercial products should be fairly straightforward, though I don't know how well they would protect in this application.
The electrolysis would do nothing to algae or critters. You should use something which doesn't contain chlorine to keep it sterile. Chlorine ions greatly weaken copper's ability to passivate. You might try polyhexanide instead. Actually, you probably won't need anything if enough of it is copper. Copper and its alloys are thoroughly antimicrobial. Hospitals really should replace as many handles and what not with copper or brass ones as soon as possible imo.
https://en.wikipedia.org/wiki/Antimicrobial_properties_of_copperYou would need in the vicinity of a volt and the amperage can be very very low as copper corrodes slowly anyway.
A major difficulty is that if you are using AC you need to transform it and rectify it in a safe manner.
I would recommend using a transformer that keeps the output winding physically separated from the other.
It is quite rare but sometimes the thin electrical insulation in transformer wire can fail and if the output winding is in physical contact you could get anything up to mains voltage between your electrodes.
Normally this just fries whatever device the transformer is powering but here it would instead just cause vigorous electrolysis which would be an explosion hazard in addition to just being dangerous from the voltage.
You might be able to get away with just low enough amperage fuse, but keep in mind I am not an electrician. I just want you to be aware of some safety issues when designing something connected to AC mains. Simply using one of those black wall wart AC adapters without due consideration is not recommended.
A potentially easier design would be to use passive galvanic protection and replace the graphite with a reactive metal. However this would require periodic replacement and it would be difficult to tune the current.
A more flashy design which might fit the theme would be to have the flowing water power a small generator which you used to cathodically protect everything.