Hard time or stalling when trying to start

I've noticed over the past few months (live in Alaska, park in a garage) that my 2010 trd has progressively had a harder and harder time starting.

Eventually, it got to the point where I had to jump it one night. I replaced the battery, that didn't work, new alternator, still have the same problem. On the first turn of the key it will start and stall after a second. On the next it will crank and eventually start.

I'm out of town for the week so wont be able to do anything until this weekend.

Any thoughts??

(Additionally, there is a random yellow/green wire with a male fitting that is near the accelerator pedal and not plugged in, any ideas as to what this goes to.

All help is greatly appreciated.