Thought these were some good questions to get us going in the right direction so i thought i'd give them a whirl
What existing Pokémon individually affect the usage of entire types?
There are actually quite a few when you think about it. The obvious ones would be all of the weather starters and dragons and steels since they effect each other directly.
Rapid spin users (forretress, starmie, donphan, tentacruel) and spin blockers also have a pretty big impact on this. Spinners allow more flying(tornadus, salamence, d-nite/ bug(volcarona)/ fire(ninetales, victini)/ ice(kyurem, kyurem-b) types to be used. This in turn brings ghosts(jellicent, gengar) into play to block rapid spin, and then in turn a team that wants to get their hazards off the field will try to pursuit trap the ghost type by using a dark type(ttar, weavile) or scizor.
Rain in general being such a big threat forces you to run something along the lines of sp. def celebi, jellicent, gastrodon, sp. def rotom-w, or sp. def jirachi. Although it doesn't bring out a specific typing per se you basically need to have a water resist or something with high special defense.
Many other pokemon require attention while building teams but few are only dealt with by a specific typing. Usually abilities/ move pool/ stats outweigh typing in terms of taking on threats defensively.
What ways are there to change a typing’s usage outside of simply countering/being countered by them?
As myself and others have alluded to previously the best way to do this is to make an opposing play style more viable. This is really the only way you can change the metagame without having a centralizing threat imo.
The genesect/dugtrio offensive core used in late BW/ early BW2 could easily fit onto a sun team and take out its main counters in the lati twins and heatran. This made rain less viable and sun more viable, kind of evening out their usage during that short period of time. I don't have statistics but from my experience in the upper portion of the ladder during that time I would say this is true. In this case this sort of was done by the countering/ being countered method but my point is that this could be accomplished by a support mon too. An example of this in the current metagame would be xatu, who can keep hazards off the field, wall a nice amount of threats and do great damage to steel types with sun boosted heat wave. I think without xatu sun teams would be much less effective since all other spinners are essentially dead weight under the sun.
What elements of a Pokémon are most important when dictating how it interacts with other types?
There are really infinite ways pokemon interact with each other so it is a very broad combination of things. This is why we CANNOT base our direction on typing alone. I'm aware the mods and TL have recognized this but most of the posts on here are about typing and it should stop. Sorry about that just got on my nerves a little.
Anywho, as I said things like movepool and base stats are probably the most influential, even more so in some cases than typing. Dragons are a perfect example because they really don't have the best typing yet they are insanely powerful thanks to stats and move pool so we assume they have the best typing. (granted they get some amazing resistances but I think its easily debatable that there are better types).
In this metagame, a pokemon MUST be able to interact well with a certain weather or counter a certain weather to see usage. That is a large reason why things such as poison, ice, normal etc. have such low usage and why water, dragon, grass, electric and fire all have good usage. In a weatherless metagame I would predict water, grass, electric and fire to all drop tremendously in usage due to the boost they are given by weather. I would even say psychic types would fall under this umbrella but they are given the best abilities in the game so that saves them.