The Gemara in Sukkah (I believe in the daf is in the high-teens) discusses a halachic principle of dofen akuma (a bent wall). The case is slightly different from your scenario, but I think still applies. The Gemara's case is when there is a bit of roofing that is not kosher for skhakh separating the walls and the kosher skhakh. Normally the maximum distance between a wall and the skhakh is 3 tefachim. If the distance is greater than that, the sukkah is pasul. But if you have some non-kosher roofing material connecting the wall to the skhakh, it is considered as if that roofing material is actually part of the wall (so the wall goes vertically up to the roof and then horizontally to the skhakh).
In that case, the sukkah is kosher; however, those inhabiting the sukkah must sit directly under the kosher skhakh. One does not fulfill his obligation for dwelling in the sukkah sitting under the "bent wall."
I think this clearly applies to your case where we don't have to go through any mental gymnastics to explain that the invalid roofing is actually a horizontal wall. In your case, we are talking about actually sitting under a wall and the halakha should apply the same. The sukkah is kosher, but one must be careful to sit under the actual skhakh.
As far as the minimal area, the gemara says on daf 2b that the minimal area of a sukkah is large enough to fit one's head and most of his body. Rashi repeatedly mentions that a sukkah is called a sukkah in the first place because of the skhakh and therefore implies that the skhakh is the ikkar of the sukkah. Therefore, I think that the minimum area of the skhakh would be large enough to fit one's head and most of his body.