Does publishing a paper with a non-edu email influence the review process? I saw many good papers on good journals (by Springer, Elsevier, IEEE, etc.) with a Gmail as a corresponding email.

I'm about to submit a paper and since I don't have (yet) an academic email, I would like to know how bad is it to publish a paper with a Gmail account. Does it have influence on the author's reputation or the reviewing process?
None. Outside/commercial email addresses are more likely to be permanent than academic ones, since people change jobs all the time. I think that a Gmail or Yahoo address is fine, and I rarely look at them when reviewing. Some venues use double-blind reviewing, so the reviewers won't know your email address or affiliation anyway.