"yo haría lo imposible por ti" (lo is after hacer) but "ellos lo harían sin problema" (lo is before hacer). Why does lo move around? Does it matter?

I think the answer is that these are different uses of "lo". In "lo imposible" it's acting as a neutral article ("the"), but in the second example it's an object pronoun ("it"). Object pronouns have to be placed before a conjugated verb. 

that's a nice crear explanation. Thank you!!