will take a stab at answering this based on comments & culture here & sketch something out that can be used as a point for further discussion because its a direction/approach Ive been thinking about & looking into & even veering towards myself & its key to this group.
this question is probably a little too hypothetical. in principle you could build a paper by stringing together some stackexchange answers but it would be unusual. as mod kaveh writes, each question must stand on its own separate merit and also note, few people regularly post questions here or with high frequency esp those with high rep, and there is some possibility that frequent questions are actually informally frowned on via the voting mechanisms & current collective voting tendencies.
the pattern is more in general that high rep users answer random newcomer questions. also, it is unlikely that you would get solid answers to all the separate (presumably nontrivial) questions.
basically yes, that is part of the raison d'etre of the group, to occasionally collaboratively generate scientific research to theoretical computer science in general & complexity theory in particular (complexity theory entries even were once highlighted here in css formatting). however the conventions around this are new & still being established as we speak.
1st, use the stackexchange search engine highly extensively and also the tag system, both can be very effective. find every relevant post to your paper or subject. figure out the best knowledge that is posted on the group. figure out the border between the known and unknown and what is posted vs not posted. this also requires close familiarity with nearby papers. it helps to bookmark or "favorite" all related posts on here & elsewhere. use the related posts column on the right also. mix this with heavy cross use of google. in some cases you may find an expert on here posting on the subj. you can look at their own answers & look at their papers.
next, try posting a single carefully crafted question totally centered on your subject and your paper and what you're thinking of establishing. its a delicate mix between the general (background) and the specific (the new result/concept you're aiming for). as if you have a single chance to ask the experts what you'd most want to know but what you also most think they'd be able to answer.
responses will help you figure out what is known by the crowd but also how relevant it is to this group based on reaction in form of comments, enthusiasm, voting, etc. it has to be at a "just right difficulty" between too easy [not research] and too hard [you wont get replies/answers]. it takes some time to find that balance & its an ongoing challenge. dont assume that if there is a good answer, someone will post it. dont assume that if people post answers, those are the best possible.
you will prob find if you are very focused that there are two gaps. one is the gap between what you want to establish in your paper and what is established in related literature. the other gap is what you can actually find out by using this group (both by searching it and posting questions). these two gaps are interrelated. you can judge whether you can build on all this with meaningful new results. but those key results will basically have to be generated by you, nobody else, unless you have found offline collaborators in your own professional circles. you can use other results as a "jumping off" point.
so finally, leverage everything youve found and write a paper. if the answers on stackexchange are very notable and innovative, you can cite them in your paper using the stackexchange urls/"cite" feature. this is really not much different than citing blogs, which is occasionally done in real scientific papers. however note that this is somewhat problematic because content on here can disappear very easily due to voting, moderators, users [who can delete stuff] etc...
probably the best and most successful model of all this invented so far is polymath with a difficult problem and published arxiv paper to its credit but keep in mind this is a project started by a field medalist Gowers and promoted by his friends eg Tao and Kalai who are also very elite/accomplished theorists.
 How will you cite a discussion on this site in your paper?
 How can we collaboratively investigate open problems?
 Massively collaborative mathematics/Polymath
 Highlights of cstheory.stackexchange.com
 What interesting open problems have been asked on this stackexchange?
 Attacking Open Questions
 References to cstheory in the literature
 Original proofs generated on the parent site
 Questions answered. in theory.
(early tcs.se ad & highlights reel for the beta period by mod Suresh)
 How to ask a good question