To go back to your basic question about joining “Deep Core Mining Inc.”:
You can NOT join it. It is a NPC Corporation.
“Whenever characters leave a (Player) corporation without joining a new one, they will become member of a specific default NPC corporation. This NPC corp will depend entirely on your choice of bloodline during character creation.” (Eve Uni Wiki)
Your toon will allway be transfered back to "Science and Trade Institute ".
Now, You will have to make a Choice:
Stay in you original NPC corp,
Join a Player Corp,
Create your own Corporation.