What types are common in the current metagame? Why?
Dragon, Fighting, Steel, and Water. Dragon and Fighting are the offensive powerhouses, with Dragon having incredible neutral coverage, and Fighting hitting a ton of stuff super-effectively. Steel, meanwhile, is a defensive powerhouse that has 12 resistances (including an immunity). Water, however, differs from the other three types by being so extraordinarily flexible, being able to work both offensively and defensively, not to mention having rain on their side. I, for one, probably think that Water is (or should be) the primary target for this CAP.
What shortcomings do common types have?
...not many. Fighting types have the most obvious weaknesses; due to their affinity to be on the physical side, they are easily stopped by a physically bulky Pokemon or a decent special attack. Steel types are powerful defensively, but much of the time, their offenses leave much to be desires, especially since the type is resisted by so much. Dragon and Water, on the other hand, are MUCH harder to find shortcomings for. Offensively, Dragon is stopped only by Steel, and defensively, it falls only to Ice and itself, though Dragon-types are often 4X weak to the former, plus they're rarely defensively inclined. Water is only resisted by itself, Grass, and Dragon, and is only hit super-effectively by Grass and Electric, though Water-Ground is common, nullifying.
What types are uncommon in the current metagame? Why?
A LOT. Grass, Bug, Fire, Poison, Ice, NORMAL (probably a non-factor now), and probably Electric as well. Most of these falter against the popular Steel type (while Poison is completely ineffective), and Fire (as well as Electric, if we're counting that), falls to the omnipresent Earthquake and is resisted by Dragon. Ice, while awesome offensively, sucks defensively, hit super-effectively by Fighting and, of all things, STEEL. Poison also suffers from Ground weakness, and sucks offensively. Bug, Fire, and Ice all fall to Stealth Rock.
Flying is an honorable mention because while it isn't exactly uncommon, it isn't very well utilized.
What positive features do these underused types possess?
Not a lot when grouped together. However, they have some good qualities individually. Both Fire and Ice are offensively powerful, being able to hit Steel and Dragon respectively. Grass types possess useful resistances to Water and Ground, as well as hitting them (and Rock) super-effectively, while Poison is also useful defensively, resisting Fighting and having few weaknesses (though the fact that one of these weaknesses is Ground is troublesome). Bug is able to take down Psychic and Dark types.
What existing Pokémon individually affect the usage of entire types?
Politoed is a huge one, as it can summon rain, which neuters Fighting-types while giving rise to Water-types and sheltering Steel types from one of their few weaknesses (or in some cases, their ONLY weakness). Tyranitar and Hippowdon can summon Sand, which increases the usage of Steel-types, which are not damaged by Sand. Ninetales summons sun, which, while doing SOMETHING to help Fire-types, destroys Ice types.
However, there's a ubiquitous factor that ISN'T a Pokemon, but is very important to the dominance of the overused types: STEALTH ROCK. This ONE move, easily learnable by many Pokemon, has neutered Bug, Fire, Flying, and Ice Pokemon, taking 1/4 HP off if they aren't neutral to Rock, severely limiting the selection.
What ways are there to change a typing’s usage outside of simply countering/being countered by them?
Being able to assist underused types, while being assisted by said underused types, is a good way to increase a types usage. Dragons appreciate Fighting types to be able to take care of Steel types. So why can't we use that for our CAPmon? It would be able to take care of Pokemon that are hindering the use of the underused types, and the underused types would be able to help it.
Finally, what elements of a Pokémon are most important when dictating how it interacts with other types?
I'd say the CAP's typing is the most important element for how it interacts with an individual type. Abilities come second, as they provide resistances or immunities (though the only useful one in this CAP would be Water Absorb). Stats can be important, as a bulky Pokemon can sponge neutral attacks, and a offensive powerhouse can deal devastating hits to those that don't resist. Movepool is most important for a large variety of types; a wide movepool keeps some usually hard-counters at bay, while Rapid Spin makes life easier for Fire, Ice, Bug, and Flying types.
tl;dr: Thus, I think the best Pokemon for this CAP would be one that can be not only a counter to those that hinder some types, but one that can assist, and be assisted by, Pokemon of underused types.