So there’s several inaccuracies in what you have listed. Here’s the proper way to do it:
Figure out the base damage type(s); Sabertooth attack is 18 physical as an example while Flurry Swipe is 30 physical.
Add any untyped damage, modified by the added damage effectiveness, of the same type(s) of damage as the base. If you have +16 minion melee damage, and Sabertooth doesn’t have Ice Tiger, then the normal attack adds 16 physical damage and Flurry Swipes with 200% added damage effectiveness adds 32. If you had converted, it would be cold damage added instead, but of the same amounts either way.
Add any typed damage, of the type it specifies, also modified by added damage effectiveness. If you have +20 cold damage from frost fang and +10 physical damage from the passive, those get added to the normal melee attack, and 2x that (thanks to 200% added damage effectiveness) would get added to each attack from flurry swipes.
Once you have taken care of all added damage, then you multiply by increased damage sources, all added together.
As you can tell from your minion tab, you have a total of roughly 300% increase damage. From passives, equipment, idols, all of it gets lumped together. Then, the total increased damage amounts applicable to the damage of a given type gets multiplied by that damage.
So yeah, melee damage is different from spell damage, and if your minion uses a spell (like a skeleton mage, sabertooth doesn’t have any that I know of) added or increased minion melee damage won’t apply to it. If it uses a bow attack, it won’t apply either. In all cases you can tell what applies from the scaling tags - note how the minion scaling tags are melee and physical and area. And you’re kinda correct in that 400% increased minion fire damage and nothing for physical would make the physical damage relatively low impact, but as you can see from yourself that would never happen. You have a lot of typed damage idols and yet it’s 350% physical and 300% cold.
So going back to the example, you have Ice Tiger, +20 cold damage from frost fang, +10 physical from the passives, and now you’re looking at increased damage; the total 38 cold damage is x4 (100% normal + 300% increased minion melee cold damage) for 152 damage, and the +10 physical is x4.5 (100% normal + 350% increased minion melee physical damage) for 45 damage, meaning total damage is 197.
Then you multiply by more damage; so if you had heavy paws that’s 40% more damage that’s (197 * 1.4) 275 damage, and if you also have sabermaw, since each more damage source is multiplied separately, it’s then 275 * 1.3 = 358 damage. This is before worrying about resistances, area level, or any of a number of other things, but relationally this holds true regardless.
The main point is, yes, increased damage can modify the effectiveness of added damage of a given type, but all added damage is going to be multiplied by increased damage and more damage and so forth, it’s just that the pool of increased damage or even more damage bonuses which apply to it might be different depending on what type of damage it is, and thus what “scaling tags” apply to it. And as far as what damage type it does, just look at the scaling tags. Sabertooth base scaling tag is physical, so it does physical damage base. If you take ice tiger, in addition to swapping that base damage to cold, it swaps the scaling tag. If ever it’s not clear, those tags should help clear it up.
Does that make a bit more sense?