We have made the first map of CO(J=3-2) emission covering the disk of the edge-on galaxy, NGC~4631, which is known for its spectacular gaseous halo. The strongest emission, which we model with a Gaussian ring,occurs within a radius of 5 kpc. Weaker disk emission is detected out to radii of 12 kpc, the most extensive molecular component yet seen in this galaxy. From comparisons with infrared data, we find that CO(J=3-2) emission more closely follows the hot dust component, rather than the cold dust,consistent with it being a good tracer of star formation. The first maps of $R_{3-2/1-0}$, H$_2$ mass surface density and SFE have been made for the inner 2.4 kpc radius region. Only 20% of the SF occurs in this region and excitation conditions are typical of galaxy disks, rather than of central starbursts. The SFE suggests long gas consumption timescales ($>$ $10^9$ yr). The velocity field is dominated by a steeply rising rotation curve in the region of the central molecular ring followed by a flatter curve in the disk. A very steep gradient in the rotation curve is observed at the nucleus, providing the first evidence for a central concentration of mass: M$_{dyn},=,5,times,10^7$ M$_odot$ within a radius of 282 pc. The velocity field shows anomalous features indicating the presence of molecular outflows; one of them is associated with a previously observed CO(J=1-0) expanding shell. Consistent with these outflows is the presence of a thick ($z$ up to $1.4$ kpc) CO(J=3-2) disk. We suggest that the interaction between NGC~4631 and its companion(s) has agitated the disk and also initiated star formation which was likely higher in the past than it is now. These may be necessary conditions for seeing prominent halos.
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