Yea I think a push is kinda bad here, mostly because it doesnt matter how big of a hand you rep if opponents range is mostly set/str8/air. I mean if you EVER get him to fold a 2pair thats a coup for you but push is still bad.

River C/R is veeeery read dependant, if you think villain will vbet river with a set with a 4str8 out there (and keep in mind he may have a higher str8 even if u hit) but not call a vbet and especially if 1,2 is true and he will call a C/R with his set then by all means C/R. Same with the flush.
However almost always I would just call turn and vbet river as unfortunately u r representing exactly what u have-- a big draw. Many villains will check river here with a set and maybe even worse will vbet 1/3-1/2 pot folding to a C/R which is imo the WORST as your checkraise riv plan works and you still lose value when compared to vbetting 2/3-psb on river.