The hole above the battery that you zoom in on at one point is for the battery hold-down bracket. If I recall correctly, the ground cable only attaches to the block and/or starter in my cars, but it won't hurt to add an additional ground since your hold down is most likely gone.
The running issue is hard to id from that video. Have you pulled codes from the ECM? There is no check engine light in the Z31, but you likely have a code or two stored in there.
http://z31.com/wiki/doku.php?id=repairs:codes