What does “adorkable” mean? How popular is this word? To what kind of objects and occasions can I apply “adorkable”?
Solution 1:
Adorkable is a fairly recent (on my scale of years) coinage which, like your Kawai appears to be most current among adolescent girls. It is exactly the sort of term which won't show up in dictionaries until it has evaporated.
Your best source for words of this sort is Urban Dictionary, a user-created dictionary of contemporary slang. It's very spotty in quality and often egregiously offensive, because most of its contributors are teenagers; but it's a model for the value of crowd-sourcing in matters on which its particular crowd is the ultimate authority. UD in fact has 9 pages on this word.
You will see there that it is a portmanteau word which combines, as you suspected, adorable and dorky.
Adorable in colloquial usage no longer means "worthy of adoration or worship* but exceedingly cute—very like what I understand Kawai to mean.
Dork is a little problematic—you may consult UD itself for its range of meanings—but all its senses seem to include a combination of intelligence, eccentricity, and social ineptness.
It appears to be applied mostly to young men by young women, and occasionally to young women by young men. Men of our years should probably avoid using the word adorkable at all.
Solution 2:
adorkable:
The word "Adorkable" is a portmanteau of "Adorable" and "Dork." That neatly encapsulates this trope. Adorkable characters are "dorky" in some fashion. Maybe they're socially inept or shy. Maybe they're really clumsy. Maybe they have some really, really conspicuous character tic that tends to earn them weird looks. Maybe they're just so darn sweet, that it borders on embarrassing. Heck, maybe they're just an out-and-out Nerd. However, rather than making them an outcast, these quirks give the character an endearing vulnerability.
There is a longer explanation of the term and there are many examples of this usage at the link.
I don't think it's the English equivalent of kawai: cute is. Hello Kitty is its archetype and primary avatar. We use almost the same word as kawai in Chinese in Taiwan. Same meaning and feeling.