Vehicle info: 2011 Jeep Grand Cherokee 5.7L V8 with 245000-ish miles on it. Laredo or overland, not too sure of trim but probably doesn’t matter anyway.

Details:

Ok so sometimes when I go to start my vehicle in colder weather it will either try to start indefinitely or start and then immediately die afterwards (1-2 secs later), the latter has happened maybe 2-3x as much as the former. What I’m about to describe next has only happened once so far, but one pretty damn cold morning I was driving for a few miles and then came to a stop light. The light turned green and I went to accelerate but nothing happened, only for me to realize the damn engine has shut off (I noticed no hot air shortly after coming to a stop but didn’t think anything of it). I just put vehicle in park, started it again, and didn’t experience any issues for the remainder of the trip. Most recently, the vehicle started and died twice in succession, but after putting car in ignition mode for a few sec it started fine.

Some more details that are probably relevant/related:

• This only happens when temperature is around or below 50F (10C). Usually happens in the 40-50-ish F range.
• The colder it is, the more violent and slow the start is (though I believe this is normal to a certain degree).
• So far, this hasn’t happened when putting the car in ignition mode for several seconds before attempting to start.
• No matter the temperature, when idling, the car vibrates and the intensity of vibrations changes seemingly spontaneously. When most intense, it’s not enough to make me go “oh shit! wth was that”, but is still enough to feel, probably enough to put ripples in your drink or smth. Intervals range from a couple to several seconds. (Idle is 500-600 rpm)

Looking online, this looks like a battery and/or fuel system issue. I’m leaning towards fuel system issue but the temperature criteria is leaving my noob *ss guessing. Guesses/opinions would be appreciated