The problem with lifting weights as soon as you wake up is that you've been fasting for a very long time (think about how long of a break you have between your last meal of the day and breakfast the next day). There's arguments for and against fasted AM cardio, but from everything I know/have read, I can't think of anything that would make it be beneficial or advisable to weight lift in that state.
As for diet changes, make sure all your meals are balanced. It's not bad to eat corn for a meal or a snack, but it is bad to eat JUST corn for a meal or snack. Try to get a good protein source, an EFA (essential fatty acid), and some carbohydrates (mix these up throughout the day, get complex fibrous and starchy carbs) in every meal. Eat every two to three hours to keep your metabolism going throughout the day.
For your questions:
1) Cardio after your workout. A short HIIT session after a workout will help you shred the fat very quickly.
2) You can generally do two muscles per day. One popular split, and one that I personally do is chest/triceps, back/biceps, legs, shoulders/traps. However, since you're just starting to work out, you may actually get better results by just alternating lower body, upper body. Alternatively, in a book I have by Tom Venuto, he suggests the following two day split:
Seated overhead press
Dumbbell lateral raises
Leg press or squats
Pullups (although if you can't do those, lat pulldowns work just fine)
Seated cable rows
Alternating dumbbell curls
Standing calf raises
Seated calf raises
All of them are 3 sets of 8-12 reps. It's a fairly easy workout, and since you don't really have much muscle right now, you probably would see more results from something like this than something focusing on very specific muscle groups.
Monday - Workout 1
Tuesday - Workout 2
Wednesday - Off
Thursday - Workout 1
Friday - Off
Saturday - Workout 2
Sunday - Off
If you want to "build your own workout", then generally speaking, some major exercises:
Quads: Squats, leg press machine, lunges (dumbbell or barbell), hack squat machine, leg extension.
Hamstrings: Lying leg curls, stiff legged deadlifts (be very cautious with these if you're just beginning, major potential for injury if your form is bad!), seated leg curl, single leg curls, hyperextension (also works lower back).
Calves: Standing/seated calf raises, calf press on a leg press machine (again, be cautious, too much weight and you can get seriously hurt). There are donkey calf raises as well, but most gyms typically don't have a machine to perform these on.
Abs: Crunch, reverse crunch, leg lifts, side crunches.
Back: Barbell/dumbbell row, chin/pull ups, lat pulldowns, cable row.
Chest: Barbell/dumbbell bench press, any sort of fly, dips while leaning the top half of your body forward, cable crossovers.
Shoulders: Military press, dumbbell press, dumbbell lateral raises, rear lateral raises, front raises.
Biceps: Barbell/dumbbell curls, preacher curls, concentration curls.
Triceps: Close grip bench press, tricep pushdown, dips with body straight, french press.
Forearms: Wrist curls, reverse wrist curls, reverse curls, hammer curls.
Pick about 2-3 exercises for any group of muscles, but keep in mind that most exercises that work your quads also work your hamstrings, so you may want to keep the hamstring exercises to 2. Anyone feel free to chime in if there's anything that should be changed here.
3) You could do a four minute jog to start, then 30 seconds sprinting as hard as you can, 30 seconds jog, repeat that three more times, then end with a four minute jog. As weeks go by, add in more of those 30 second sprint / 30 second jog "reps" slowly. Alternatively there is a 20 / 10 version that Guejsn recommends that will work just as well.
As for your motivation...don't sweat it. Get a journal and track your progress. Tell yourself "hey, you know what, tomorrow is just the day I go up one rep, and that's better than yesterday". If that improvement isn't enough to keep you motivated, try to see if you can convince one of your other friends to go to the gym with you. There's always affirmations...they're a bit cheesy, but if telling yourself that you're the best every morning is what it takes to get you motivated, then more power to you. It helps to set concrete goals too, and when you achieve them, great, you get to set new goals and become a better person. Set reasonable time frames, and if you can't meet your goal by that time frame, don't see the failure as a downer. Look at what you did, see if there's anything you could change, and get back in the gym. Hell, even just getting some great music that will pump you up is an excellent way to get in the gym, I know when I hear a song I really like it gives me incredible energy. If you're working hard, eating clean, and staying determined, you WILL see results.
Best of luck brother, go tear up the gym.