Do support gems override or extend the "type" of attack skills

It sounds intuitive that when linking a support gem to a skill gem, the support gem can in a way alter the type of the skill. For example, it makes sense that linking a Ballista Totem Support to a Galvanic Arrow will make the skill benefit from effects that increase totem damage.

What's less intuitive is whether the skill altered this way still benefits from effects that boost the original type of that skill. To use the previous example with Ballista Totem Support and Galvanic Arrow, will taking passive skills that increase bow damage affect the damage of the totem as well?

The wiki page of the Ballista Totem Support says:

Totems use the player's own skills as though the player had cast them, ...

... which would imply the skill will still benefit from increased bow damage, but it doesn't directly confirm or deny it. I'm still new to the game and would really prefer a confirmation before spending my hard-earned skill points (considering how hard it is to unlearn passive skills).

To put simply, extend is the answer and yes it will work.

Support gems will not "override" the original gem type. In your specific case, the bow modifiers in your skill tree will increase the damage of your Galvanic Arrow. You are not increasing the damage that all totems in your build do, just totems that use bow skills. It's a very small distinction and may seem annoying to point out, but an important distinction to clear up.

If you took passive increases to totem damage, it would also work in your case because the skill (Galvanic Arrow) is being used by a totem.

A good way to check to see what increases work with your setup is to use something the community uses frequently called Path of Building. Make sure to use the "community fork" because the original isn't being updated anymore, so the community came together to work on it and keep it updated with new gems and items as the game grows. You can plan, update and import/export entire builds using this. You also get output of all of your characters stats in real time as you fill out the passive tree and update your items.