<What rifle makes for the best carbine?
Depends whom you are arming and for what purpose. 94% of the time the answer will be either barbiedoll AR-15 or AK-74 The Rifle Dynamics AK-74 is technically a carbine because with a 14.5" barrel, it has a shortened barrel compared the 16.1" original AK-74.
<What cartridge is best suited for a carbine?
Again, depends on what you're using it for. For a general purpose gun 5.45x39 is the best practical round, and 5.56x45 is imperfect but more than good enough if it's abundant in your area. .224 Valkyrie is objectively better than both ballistically speaking, especially when you factor in weight, but its position as an expensive, relatively uncommon meme round makes it impractical to adopt in most cases. It's also new and relatively unproven in adverse conditions.
<How do bullpups compare to traditional carbines?
At this moment, with the current lineup, they're mostly shit for general purpose but have some limited use in certain niches. That niche could be made a lot bigger and a fair bit firmer if some company went out of their way to design a bullpup rifle from the ground up instead of taking a conventional rifle, ramming the FCG forward, and lopping off the stock.