You're right. I have been thinking about this matroidally, and such conditions come for free in that setting so it didn't occur to me to include them. Your observation is right. In particular, any line L of length q+1 must meet all other lines (otherwise, any point on a line L' that does not meet L lies on q+2 lines - L' itself, and one line for each point on L), and by applying pigeonhole to points on L, we can find a point on L lying on at least q+1 lines not containing e, and therefore at least q+2 lines. So WMA that each line contains at most q points. I don't know how useful this is.