[2022] Is there any way to see the base damage of a skill?

The base damage of most melee and bow attacks is only around 2 or 3, as they're intended to feel like ways of conveying your weapon's damage.

The base damage of most spells and throwing attacks is 1/5 of their added damage effectiveness, so a spell with 100% added damage effectiveness will generally have around 20 base damage and a spell with 400% added damage effectiveness will generally have around 80 base damage. This fixed ratio means that with no other sources of flat damage, +1 flat damage will result in 5% more damage.

Voidknight spells are a bit different. Their base damage is generally 2/5 of their added damage effectiveness, so a void knight spell with 100% added damage effectiveness will generally have around 40 base damage, so +1 flat damage will result in 2.5% more damage. This, combined with their flat void damage per Vitality scaling is intended to make them less reliant on the use of specific caster weapons such as wands and staves.

edit: fixed 5% -> 2.5% in the Voidknight spell section.

Do flat increases from the skill tree itself (like Hammer Throw's Avatar of the Spire "physical damage: +12") get multiplied by the "added damage effectiveness"?

Yes

And likewise, does Void Knight's Void Bolts "Throwing Void Damage: +2 per point" get multiplied by the "added damage effectiveness"?

Yes