Will doing research with a prof at Caltech help me get into their grad school with low gpa and out of state?

Low GPA could be affected by none core engineering related subjects like history or gym (although colleges don't give gym anymore these days). If that is what is pulling down your GPA, then there should no problem here. Otherwise a low grade in a key engineering or science subject may need that to be repeated to show that you will work at it to gain competency. From you SAT scores, I don't think that is your problem.