For example, if an artist on pixiv has a reccuring character prevalent through their works, would it be bad form to tag the characters' name on Gelbooru?
Does the character/artist have to have certain popularity to deserve a tag?

If the character has a name it's ok to tag it.
Just make sure you add "original" to the tags.

I've just realised that there is a tag "original_character" but it have no description in the wiki. Some tags got that note as obsolete and even direct advice what to use instead, but here I'm confused. Since color let me think, that "original" tag is about source or copyright, and "original_character" is green so it supposed to be about character that don't comes from the universe as it was created. For example there is many original characters placed in naruto universe for example, made by pixiv artists.
And what's more to confuse me about this tag is I remember that long time ago it was more common, while now there is just 2 pictures popping on it.

For example: gelbooru.com/index.php?page=post&s=view&id=1737243