Early morning runs

you can run without eating

you can definitely run without eating.

However, depending on how far you're running you may find that having a small pre-run snack is helpful for energy levels. Try stuff like a banana and some diluted sports drink. Or toast & honey. Basically, easily-digesible carbs will do the trick.

As long as it's not a crazy-hard effort (i.e. race, or long-hard effort) I can usually get away with eating something like this 30-45 minutes before my morning run.

I run about 70 miles a week around 5-6AM and rarely eat anything. I have a glass of water, a glass of orange juice, and a cup of coffee. (Yes, I normally have all three). Liquid, carbs, caffeine. Take a nice sit-down, too.

I either eat nothing or eat a quick granola bar, although I always drink a glass of water. You don't need to eat, and actually for me I usually try not to eat close to my runs to avoid stomach cramps.

I get up, go to bathroom, brush my teeth and run. I've never had a problem. I will tell you I run slower with the same perceived effort in the AM.
I'm in my 50's, in the am the first mile is generally 8:00-8:10 and I average about 7:30-7:45 for 6 miles not really pushing it.
In the evening, I can go out in 7:45 and average 7:20 with the same effort.

Whenever I run in the morning I end up late to work/school because I'm shitting my brains out. Don't know what it is, but morning runs + coffee + orange juice/breakfast wreck havoc on my GI tract.

Glycogen stores are partially depleted when you wake up. So if you're training for the marathon, not eating may provide a stronger training stimulus for using fat. If you're doing 5k-10k kinds of workouts you'll likely be slower at them for several weeks. I like to wake up a little early and drink something with caffeine and carbs

Maybe time for new foods or a new breakfast routine?

Best to consume AT LEAST 250 CALORIES IN ORDER TO ELEVATE BLOOD GLUCOSE LEVELS--THEN GO RUN. THIS CAN BE CONSUMED IN THE FORM OF FLUIDS IF NEED BE....PERHAPS SOME WHITE BREAD WITH A BIT OF JELLY ON IT AS WELL. YOU WILL FIND THAT YOU HAVE BETTER RUNS..

clif shot bloks have worked for me, little elecrolyte gummy things, dont mess my stomach up but give me a bit more energy, and i usually drink diluted gatorade with them

Just Run Baby

I run basically every morning before heading out to work a little before 6 am. Wake up, go number 2, and head out the door. You'll start slower but on the whole will get in a good routine and learn to be more efficient with your fuel. It translates real well into marathon training for sure... I can run 20 miles hard without any gu or gel now.

I sleep in my workout clothes for the next day, so all I have to do is wake up, use the bathroom, and drink some H20 before starting my run (usually about 9 miles at 6 am). Why waste time getting ready, eating, and digesting if you're in a rush?

I find that my legs are very stiff in the morning. Will this improve as I get use to the morning runs? Also, if I am running slower, will I still have the same fitness benefit by doing morning runs as oppose to evening runs?

It might take you a mile or two to warm up completely, but there is no reason you shouldn't be able to run as fast and get the same benefit you would from evening runs.

How do you guys manage poop out everything before the run? Usually when I wake up, I try to go #2 immediately, but nothing really comes out. Fast forward 30 minutes into the run and I gotta pull a fast one in the trails or side of the road. Is there something I should be eating/drinking the night before?

Quit being a chode man, wake up have a cup of coffee, I'm sure running will seem like a good idea about 20 minutes later

you need to experiment and get into a routine that suits you best. Some runners need to shower and warm up others need snack and can get out quick. Probably start off with sports drink and energy bar but see how it goes over time.
All runners are different and have various needs to perform at their best.