i can only speak in terms of fiction writing, so take it for what it's worth.
'important' is rather relative, isn't it? i assume you mean a major character, and those usually have resolutions, too, at least to some extent. i can't think of any off-hand major characters that don't have some kind of story resolution, but i'm sure there are plenty. supporting characters, not so much (for example, sallah (indy's friend) in 'raiders of the lost arc, er, ark.' indy would have been rather screwed without sallah's help, and we learn quite a bit about sallah in the process, but he doesn't have a character arc to speak of (if he's got one at all)).
do we need to know every detail about sallah's life? not really, but being a pretty significant supporting character it's only fair that we know *something* about him even if he basically fades into the darkness when his purpose is done. the exposition about sallah is rather brilliantly done, you don't even know you're being told about this character, and in knowing him we like him, and, imo, that's the key to his humour. his character is more than just some guy's whose purpose is to move the plot along.
you wouldn't treat your major characters any differently, no? your major character may be the one with the actual character arc. you'll see sometimes where the 'main character' doesn't change from start to finish, but those who tag along or get caught up in the action are the ones who are changed by the end.
for me, and this is just me mind you, i know pretty early on what my ending is going to be. the earlier the better (for me). that way i know how the character is at the end, which tells me how they should be at the beginning. everything inbetween i consider merely (hopefully) entertaining justification for that change. that's how i personally develop a story. also, i have a corral of 'stock' characters in my mind that i variate and riff off of and that helps me know the character without doing all sorts of detailed backgrounds and outlines and such. that's just me, and those are normally just side characters i do that with (though those side characters were at one time main characters in something else. i used them up and in the process of coming up with something different, they're set aside for support. my theory there is i know the character type already, and if i thought they were good enough main characters then they're certainly good enough support cast. but, out with the old, in with the new).
so, yeah, imo major characters and even certain side characters needs, or could stand to have in a lot of cases, their own thing going on. the caveat: i can't tell you how many times i've seen this written; 'my side character is more interesting than my main character! help! what do i do?' if this isn't a good enough reason to do an outline, i don't know what is.
i can tell already that my process for screenwriting will be a lot different than that for fiction. in fiction, for stories i'm serious about, i find myself retro-writing a lot of it. but, being a hack, i'm confident that the story will be the way it 'should' be. it might suck, but it sucks the way it 'should' suck. i take a lot of things i do in fiction for granted, but i'll have to consciously be aware of for a script. i'm fortunate in that i tend to pretend to know what's 'supposed' to come next, so that there's hopefully an organic kind of development to it. i think of it in terms of 'what should happen naturally next given the situation?'
sorry, it's hard to answer 'how do i advance my plot/end the story with what i've got?' without really knowing what's going on with the story. that's when i might start from the end and work backwards, thinking of the scenes that leads me to the place i'm stuck. i can't say it's a lack of conflict when it could be a lack of anything. it could even be that the character has peaked too early, satisfying his arc before the end. it could be this, it could be that, it's impossible to say.
