Alright, there is a lot of stuff going on here, lets take it piece by piece:
1 - The diet is terrible. I know you dont want to be a body builder and are looking mainly the lose weight, but you gotta get some protein in there for multiple reasons. You will lose weight on that diet for sure, but when all is said and done you could potentially end up "skinny fat" where you have no muscle to be seen, and just happen to weight 50 pounds less than you do right now.
As much as possible you want to be eating things like eggs, egg whites, chicken, protein powder, and lean beef for protein sources. A protein source should be at EVERY feeding. Carbs should be rice, veggies, potatoes, yams, fruits, yogurt. And dont forget fats: Olive oil, egg yolks, avocado, cheeses, nuts. I dont think you need to be killing yourself eating 6 times a day if you a beginner. I would focus more of getting 3-4 complete meals in. A banana or carrots is not a meal.
Example meals would be: 3 eggs, cheese, avocado, turkey, or 1/3 pound 96% lean beef, avocado, cheese, veggies. 1-2 scoops of protein with peanut butter in 1 cup milk, blend with ice.
I'm biased towards low carb diets because thats what works for me, so you'll have to play around and see if you can get away with Chicken and rice, or Small steak and yams.
2 - I guess given your minimal set up and current situation the workout isnt too bad. Although the key to getting better results is the progress, so make sure you are getting more reps or adding weight to the bar... Even while dieting if you have been out of the game for a while. If you are looking for more cardio style stuff, Burpees are good as is jump rope, and both only need a small space to perform and are VERY effective.
I work from 12-12:30 3 days a week, so I know what its like coming home at 1:00 and needing to work out still. Unfortunately all I can say is just get it done. If you really want it bad enough you will, and thats all there is to it.