I started to exercise more lately and I've noticed two things:

First, the more I exercise and work out, the hungrier I am. How the hell am I supposed to lose weight, and trim down, if I'm literally binging on high calorie, high carbohydrate foods all day long?

After running 3-4 miles a day (usually around 8pm), I find myself craving carbohydrates 2-3 hours later (around 11pm). Coming from someone with strong willpower, It's seriously uncontrollable. I start gorging myself on complete crap: enriched white flower tortillas which I make homemade burritos with, and vanilla ice cream with peanut butter. I can't stop myself from eathing the junk, yet I feel like complete crap once it's down the hatch.

Second: While running I get these really strong cramps in my stomach; shooting down both sides of my abdomen. Short of stopping mid-run, Is there anything I can do to minimize them or make them disappear?

I supposed to loose weight

No, loose weight is a bad thing.

I start gorging myself on complete crap:

Same here.  It's almost like a survival instinct that you can't fight.  I found that giving-in much earlier and eating a little helps a lot.  It's better to eat, as in your case, a few potato chips at 9:30PM rather than eating a lot more at 11PM.  I also found that keeping my carbohydrates below 25g per day gets rid of this craving completely.  The only problem is that you have to work very hard to do that.

While running I get these really strong cramps in my stomach.

Breathing slower and more deeply is the only thing I've found that will help.  Other than that, just keep running.  For most people that problem goes away relatively quickly.z

Are you taking/eating anything after your run?  It's important that you do take some carbs (and protein) in right after the run, and if you aren't doing this, it may cut down on your later cravings.

1st question - how good or bad is your physical condition?

Sub questions - how old are you?
                        - how fast are you running?
                        - how far or how much time are you running?
                        - what is your height/weight
                        - are the cramps related to the intensity of your exercise?  when do they begin to occur in your workout?

Your binge eating is most likely force of habit.  When sedentary people first begin to exercise, they will experience increased appetite.  Work around this by eating healthy snacks and eating smaller meals/snacks more frequently.  Eat an apple (100 cal) or a couple hard boiled eggs (75 cal apiece) to satisfy your late night cravings.