I do mine about the same way, except I also cross the straps (strap hooked to left side of axle is attached to right side of trailer). I've never attached anything to the frame and had no problems after several thousand miles of towing my K5.
For the hooks that attach to the trailer, I would run them through the center of the stake pockets and then up and over the edge, actually hooking them on the top edge of the pocket. My trailer has d-rings on the deck and I use twisted snap hook straps......it would be almost possible for one to come unhooked and the d-rings are easier to hook up and prevent having to run a strap or chain over a sharp edge that could cause binding or a cut on the strap.
My few regarding attaching to the frame is that it's difficult to compress the suspension enough so that it will not further compress when hitting a big bump. If it did compress more, it would slack the straps/chains and could cause them to come unhooked or simply break due to the shock load. Commercial car haulers do attach to the frame, but they have a setup that is capable of pretty much compressing the suspension to the bumpstops.......you can't do that with normal ratchet straps!
However, running some additional straps to the frame as a safety if the axle straps break probably isn't a bad idea. It may also be necessary if you have a really soft suspension on the trailered rig in order to avoid excessive sway.