For the 1x3, I screwed the first one through the sides of the cabinet and into the 1x3, using two screws per side. On the last cabinet, the screws are underneath the cover panel on the end. I installed the cover panel before I put the Island Support System on because I had to notch the cabinet and cover panel on the end, and I wanted to notch it all at once (I used a multitool for the rough cutout and finished with a router). But you could wait and notch the cover panel separately, as I did with the long back cover panel.
On the next cabinet, I staggered the 1x3 to put it back slightly so I could get one screw into the side, which you can see in one of the photos. (if they were lined up exactly, you would have to put the screw in at a pretty severe angle). For the second screw, I had to use a small L bracket under the 1x3.
I suppose you could use L brackets in lieu of screwing through the cabinets. I can't recall exactly what I used, but I think it might have been these, 1.5 inch brackets, or maybe the smaller 3/4 inch ones.
The structural connection for the 1x3--both on the cabinet and the support--is not super critical because it only holds the support in place, so the little L bracket is enough. Once there is a countertop on the support, the countertop holds it in place, very little strain is on the 1x3. Instead, the strain is on the lever point at the rear of the cabinet where you installed the 2x6 or 2x8 (for the Hidden Island support, you could use a 2x4 back there, I just wanted to be able to put more screws into it)
For the base, I screwed through the bottom of the cabinet (particle board) into the 2x4's. Since I was using drawers, none of the screws will show, so used a lot, like 8-10 per cabinet. I think I used a combination of #8 and #10 screws, probably 2.5-inch or 3-inch. I used more on the front, because any leverage that is put on the overhang will act on those screws.
Q