~and that what this all really means is that I am a failure in life, and that I can't do anything right. In other words, I ultimately feel like I'm doomed personally and professionally.~

this seems like depression is distorting your vision of self. that's why you can't see who you really are. you seem to be doing a lot of positive things in your life, school, being independent, etc. don't lose sight of these.

another thought we sometimes revisit "false beliefs" about ourselves. like "i can't keep a job so i'm worthless." this setback has thrown you for a loop. try to see your job situation as a setback but one you can resolve. it's a temporary situation. i'm glad you have a place to live over the summer. one less thing you have to worry about as you turn things around. try to find something over the summer to replace your jobs.
this article and video may help you:
http://jonathanbudd.com/false-beliefs/

from what you posted i can tell you are a very responsible person and have been positively creative to continue on your self improvement in life. i'm sure it will all work out. you have perseverance. keep us posted okay? we're here for you.