I should have said you can't assume all immigrants are homophobic or hate women.    
As to background checks, they should be thorough.   But how do you discover prejudice if there is no record of discrimination?   Do we not already have  homophobia and mysogeny with American citizens?   Should we deport them?

I do support tough screening but I think that's being done with what resources we have.

No screening will be perfect in stopping a person with motivation, money and connections.      If we could do that,!we'd be the police state you fear.