America still provides access to the American Dream to all people. How can I defend, challenge, or qualify this statement by using ethos, pathos, and logos?

There are several things you will want to consider in completing this assignment. The first is to understand how the specific rhetorical tools you need to use are applicable to the issue.
Proof from ethos is an argument based on the character of the speaker. It can be of two kinds. Intrinsic ethos is inherent in the way you speak and present yourself. This would include using sound reasoning and correct grammar to show yourself to be educated and reliable and making thoughtful and balanced assessments rather than exaggerating. Extrinsic ethos is actual personal history or credentials. Here, you could discuss whether your parents achieved the American Dream and how and then talk about your own prospects for doing so.
Proof from logos means proof by means of argument. This is quite different from "pragmata" (a Greek term meaning "facts"). As you read Aristotle's Rhetoric or your rhetoric textbook, you will find several different types of logical argument, including those from deduction (enthymemes), induction (examples or signs), and common topics such as the lesser and greater or definition. 
Your first logical point is that the claim that the American Dream is accessible to "all" people can be refuted by a single counterexample. A claim that it is still accessible to most people could be argued for or against, but a key logical point is that a universal claim (one that uses such terms as "all") only needs a single counter-example to be refuted. Thus you can use Nate Rawlings's article for evidence of the problems "dreamers" have in achieving the American Dream, especially now that Donald Trump has instituted so many anti-immigrant policies. A second example would be Al Szekely, as described in Ehrenreich's article. You could also look at issues of definition and the greater and lesser by thinking about whether the definition of the American Dream is a moving target. For example, forty years ago a television would have been considered a luxury and mobile phones did not exist. 
Pathos involves arousing the emotions of the audience. You could use sentimental examples to describe this. For example, in the wake of the current administration's attempts to defund many social safety nets, you could look at a family bankrupted because a parent gets cancer or a child is born with a disability. You could arouse pity for a senior citizen who no longer has access to meals on wheels. You could therefore use pathos to arouse sympathy for families who cannot participate in the American Dream due to lack of a single-payer health care system and a social safety net.