If you like. But try to avoid lengthy quotations from the reading; paraphrase whenever you reasonably can. Remember that your target audience for the paper is someone who has already read the text but doesn't understand it: You're not likely to help him come to understand it by throwing chunks of the text back at him. Finding appropriate and important piece of text is all well and good, and can be helpful if you have set the reader up to understand an important bit of an especially confusing text. But wherever you're tempted to insert a lengthy quotation, ask yourself: Is this really necessary for my paper, or am I just putting it in because typing is easier than explaining? See if you can pare the selection down a bit, interspersing short quotations with clarifying paraphrases or explanation of the point in your own words. And don't forget to cite: give proper credit to your sources!
I'm not picky on this point. You can use whatever format you like for your references, so long as it is clear what author and page you're referring to and so long as you stick to the same format throughout your paper. In most cases, it will be clear which text you're referring to (most often something from a text we've read for class). In that case, you can just use parentheses rather than a footnote. An example:
Nagel says that we are merely "spectators of our own lives" (Nagel, 25).
If you are taking History of Ancient Philosophy course, there is a special pattern of citation that you should use: quotations from our source texts should use the marginal numbers from the text (e.g., 424a). (Your handout for the paper assignment will be more specific about the details.)
If you use a quotation longer than two or three lines (and do you really need that much quotation?), you should present it in a separate, indented paragraph.
If you have made use of some text that was not assigned or recommended for this course, then you need to attach such a section. And "make use of" here extends beyond quotation: if you have used any ideas or arguments from another printed or digital source, you must make note of it in a bibliography section of your paper. If you are using only the texts that "came with" the course (i.e., were assigned or recommended by me) then even though you must clearly cite the work and indicate page numbers from which your ideas and words are drawn, you do not need to provide bibliographic material. (Note: this is advice that is specific to my courses: some other instructors will want a full bibliography no matter what.)