Sorry to hear that life is so stressful. I think you're amazing for balancing a family, studies, and your weight loss efforts at once.
It is true that many people struggle with this problem every day, but like PleaseLuise said, it certainly doesn't mean that your struggles aren't difficult! I certainly would be feeling the same way as you if I had all of that to deal with.
Even if you feel that you're dragging on your workouts, I think it's awesome that you're still doing it at all. All you can do is take it one day at a time. If you go to work out and can only make it 5 minutes, it's better than if you didn't try at all. Same with your eating- take it one meal at a time and try to make healthy choices along the way. If you slip up and have a cookie, at least you didn't spend the entire day eating junky food and making yourself feel worse.
I'm so sorry to hear about your dog

It's terrible to lose a pet. I also sympathize with the challenges of caring for an 18 month-old- I was once an AuPair (nanny) for an 18 month old girl, and it is the hardest job I've ever had. I've never been so exhausted and frustrated in my life, so I can only imagine what you're feeling with everything else on top of that.
Keep your chin up and take it one day at a time. This too shall pass. I wish you all the best!