You may include others' work (quotes, images, etc.) in your own through the principle of fair use. Academic uses often fall under fair use, and there are excellent guides to fair use at the following sites:
SHORT ANSWER: YES (as long as it's not work that's completed in the course of employment by the university).
Generally speaking, you want to retain your copyright ownership, if at all possible. Copyright ownership gives you the exclusive right to:
Most undergraduate journals allow you to keep your copyright; many professional journals ask you to give your copyright to them. Always be aware of what rights you are giving to your publisher, and ask if you are unsure. I've included a couple examples from undergrad research journals here:
Other journals may ask you to assign a Creative Commons license to your work. In this situation, you retain copyright, but you grant particular permissions for how others may use your work.