My default advice in your situation is no, don't offer to review.
Aside from your technical expertise, an important qualification for a reviewer is that they have worked in their subfield for long enough to have a broad view of existing work and the state of the art. When reviewing a paper, you want to be able to think about how it fits in with what's already been done, and how important an advance it represents. You'd want to be aware of important related work, and looking up the paper's references is not enough, because the authors themselves may not know about all the important related work.
In general, I wouldn't expect that an undergraduate would have that level of experience, and thus they wouldn't be qualified as a reviewer.
If you really feel like you do have that level of experience, then I would suggest that you next talk with your advisor and/or your coauthors, and ask for their honest opinion. A more common way to get started would be to ask if you can help with reviews that have been assigned to them. (They may need to ask the editor's permission for this.) That way, you get to see what's involved, but the final responsibility for the review would stay with your advisor.
As to your specific questions:
Editors evaluate reviewers mainly by looking at their previous publications, and deciding whether they demonstrate sufficient technical knowledge and experience in the field. However, there's also an "honor system" component: when a reviewer is invited, they're implicitly expected to honestly evaluate whether they have the qualifications to review the paper properly, and decline the invitation if they don't.
When a reviewer is sent a paper, they are expected to keep it confidential and not share it with anyone. (Some people feel that there is implicit permission to share it with the reviewer's graduate students, if the reviewer feels they are trustworthy, but that would not apply to you.) So if you are a reviewer, you should not work with anyone else, unless you ask the editor for permission to work with that specific person, and the editor agrees.
Yes, in most academic fields, the majority of reviewers are faculty. Postdocs are usually also considered qualified, and in some cases graduate students might be. In some fields you might also get a significant number of researchers from outside academia, but they would typically have a similar educational and publication record as faculty.
Yes, I think that if you do decide to review, you ought to disclose to the editors that you are an undergraduate. I think it's important information that the editor needs to know in order to evaluate your qualifications, especially since they probably don't get very many such offers from undergrads and wouldn't by default think of that possibility.
