No, rooting doesn't help with signal, data drops, etc. And YES, they will know at Verizon that it is rooted if you return it rooted.
+1 on the advice that you call Verizon if you are having signal/service problems. In TX for about 2 days, they were working on the 4G network and if I hadn't called, I wouldn't have had any idea that was the case.
+2 call verizon for signal/service issues.