Is wood strength affected after weathering wood?
I've used the steel wool + vinegar method before*. It's essentially a stain so really only affects the surface layer. It shouldn't have any meaningful affect on the strength of the piece of wood. Splitting and warping is related to moisture and drying so wouldn't really be affecting by weathering stain. 

After you weather it, note that most any additional finishes will change the color. With pine (and actual drift wood) I've found both water and oil based polys will give it a slightly darker, slightly yellow tint (varnish will typically add color as well). So definitely use test pieces before committing. 

Rule-of-thumb is that water-based finishes will add less color than oil-based finishes. That said, if the goal is to 'age' the wood, perhaps no finish at all is suitable. Let it age even more in use. :)

PS (*) one trick is to pre-'stain' the wood with black tea. This puts more organic material into the surface of the wood for the oil and vinegar mix to react to.