I cannot speak for all fields, but at least in physics, materials science and chemistry, I am not aware of any problem with a graduate student being a corresponding author.
Typically, the corresponding author indication tells the reader who can/should be contacted in case of questions, requests for data, additional information, etc, and for convenience offers the (then) current e-mail address. Moreover, in many journals, this designation is not mandatory.
Yes, I've heard opinions that the corresponding author should be either the faculty ("because the students will leave soon, but the advisor will remain, and thus her/his contact will be more prominent") or the "main" author (who is typically the first author). In my opinion, either way is fine, but not for the reasons listed. It should be the person who will be the best to communicate on behalf of all the authors (i.e. knowledgeable about the project and able to communicate).
Now, I am (and most other people responding here are) talking about "corresponding author" designation as it will be printed in the journal, which can be different from a similar designation in the article submission process (aka "submitting author").
When I was a graduate student, I designated my name as the corresponding author once or maybe a few times, - as I fell some sort of pride in that ("I've grown up enough to be able to do that!"), but then realized that it was mostly irrelevant. In the future, as a postdoc and a faculty advisor myself, I avoided this designation when possible.)
Usually, the corresponding author designation is helpful when authors from multiple groups are involved, and it is not clear for the reader who to contact. As an experienced reader myself, in the absence of that designation, I would try to contact either the first author or the advisor of the first author.
Being a graduate student (or for that matter any student) is sufficient to list your affiliation with the university. Moreover, unless there are some special circumstances, you should list your university affiliation. It would look strange if you didn't.
Designation of the corresponding author (or lack of such) is very unlikely to affect the review process and the decision. What might however raise a red flag with the reviewer or the editor is the absence of a faculty member among the authors, especially if none of the authors has a recognizable publication record in the field. While it is not prohibited, and is not a barrier for publication by itself, it might be an indication of one of the following two situations:
1. A graduate student(s) is not doing something right.
I've seen several cases when students were attempting (successfully or not) to publish without the knowledge of their advisors (sometimes including them as coauthors, sometimes excluding): in one, the graduate student was misguided by some weird believe (in part, probably due to a different cultural background) that he had to publish some paper as the sole author, without his advisor or collaborators in order to become successful and find the next position motivation). In another case (IIRC), the student wanted to submit the publication against the request of the advisor to wait, due to concerns about the validity of the data.
2. The author(s) is(are) not a bona fide researcher(s) but a person(s) with some mental problems.
(I received plenty of "preprints", proposals, etc. about new perpetuum mobile design, alternative relativity theory, etc. from such people.)
However, there are legitimate cases when even a graduate student might publish without his/her advisor. I am actually aware of the situation where a very modest advisor was refusing to put his name on the publication of his student "because of lack of significant contribution", which was actually a problem for the graduate student, who wanted to have at least one publication co-authored with his advisor before he'd graduate. (For the most of the scientific world, graduating without any publication with your advisor would look weird, and could be an indication of some problems.)
In the follow-up comment, you mentioned some other faculty member willing to critically read your article prior to submission. Depending on the actual contribution of that person that may qualify as an authorship. But I had multiple occasions when I gave my manuscript to my colleagues for comments prior to submission, and no authorship by them was expected or assumed.
Now, just in case, let me offer a related advice of caution about the authorship. I have seen situations when graduate students were not realizing what type of contributions qualify for being listed as an author. And that led to very unfortunate omission of the authors from the paper. The authors who were omitted often include people from collaborator's group (if you interacted only with one person from that group you might not be aware of other contributors from that group, including that person's advisor), sometimes people from your own group (a person might not have crunched the data, but provided a crucial for the paper insight or hypothesis).
One other aspect is if you are a visiting researcher in someone else's lab using that lab's equipment, it means that the PI of that group (the faculty) has provided you with the opportunity to conduct your research using his/her, often sufficiently unique equipment (method, technique, etc.) without which you wouldn't be able to obtain your results. Depending on the circumstances, that can also qualify for the authorship.
Finally, the issue of including advisors. Sometimes, some graduate students do not realize the contribution of their advisor to the project. I've actually heard some graduate students expressing an opinion like this: "I did all the work, I didn't ask or get any help, I wrote the paper myself, why should I include him/her as an author?"
It depends. While in some cases, you might not have to (and an honest advisor will probably tell you if that's the case), in many cases, there are several legitimate reasons why the advisor should be a co-author:
a) The advisor is providing you with the opportunity to work in his/her lab, using his/her, often highly customized, equipment, methods, tools, approaches (e.g. software libraries developed in the group), etc.
b) If any part of your research project (including your RA-ship) was relying on the grant money obtained by your advisor. (Why that matters? Besides other debatable arguments, it most likely means that your project was a part of a larger funded project which in essence [and you might not realize that] was conceived by your advisor.)
c) If at any stage of your project you received valuable suggestions related to your research (including the initial idea for the project).
It is hard to list all possible circumstances, - this is just a guide of how to evaluate this, not a comprehensive list.