There can be large storms with very high intensities; Hurricanes Allen and Gilbert were fairly substantial storms that were large in nature. I don't think there's a very high correlation between size and intensity, though there may be a slight correlation due to small sample size. Similarly, there isn't necessarily a connection between eye diameter and intensity; Isabel (for one) provides proof of this, with a very large (50nmi) diameter eye.

Size and intensity do seem to be correlated in terms of rapid development. Smaller storms can both gain and lose intensity at a much faster rate than their larger counterparts. Several different physical phenomena play a role in this; someone else can probably fill in more on this, as my mind isn't fully functional right now (somewhat tired). This is partially due to the smaller storms being more susceptible to weakening by shear and colder SSTs as well as strengthening by warm eddies in the oceans.