2010 Corolla LE Starting Issue

#1
Every once in a great while when it's cold (I live in New Hampshire) we hit an unusual situation starting the car. You turn the key and the car tries to start but won't. You release the key and it keeps turning over until you turn the key all the way to off. To get the car started you have to reengage the starter & put the accelerator all the way to the floor like a flooded carburetor car. Once you get it started this may not happen again for quite sometime.
Last time I changed the starter and though I had it. This time the starter is only a year old.
Two questions;
  1. Has anyone run into this and what was the resolution?
  2. Where is the starter relay located? (not the starter solenoid)
I want to swap out the relay and have looked in the relay/fuse box under the hood & it's not on top of the relay box near the drivers knee.
 

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