When submitting a research paper as an independent researcher, is it advisable to have a professor look at it first?
I'll try to answer you two specific questions:
“Is it advisable to have a professor from a university look at your paper before you submit it?” — Yes, if you have no previous experience of publishing papers (as a principal author). Academic publication is a game with a set of written rules and some non-written expectations. You can find the rules indicated on the publisher's webpage (guidelines to authors, editorial policy, etc.). If you have read a good number of papers, including many papers from the specific journal (or conference, or …) you are submitting to, then you may have inferred some of the non-written expectations. But otherwise, having someone to help you with the more formal part of academic writing will enhance your paper's chances.
“Is it advisable to have a professor from a university look at your paper before you submit it?” — Yes again, if you do not have a very good understanding of the field, its advances and its directions. To be somewhat brutally honest (and in line with paul’s comment): from experience, newcomers to research, or to a specific field, have a natural tendency to reïnvent the wheel or overestimate the impact and novelty of their work. It's okay, judging these correctly involves a very good grasp of the overall field, both in-breadth and in-depth.
“Would it take the journal a longer time to referee your research paper if you have no affiliation?” — No. For two articles of the same quality, with an unknown author, I don't think the presence of an affiliation would impact review time.
Edit to answer question in comment: “Would a professor who reviews the paper want credit?” — It depends on the amount of work required and the ethics of the person you ask. If reviewing the paper requires significant amount of time, especially because there are many faults with the paper that need to be fixed, you could see that there must be incentive for the professor to invest in it. In all cases, discuss this issue upfront!
The question is a bit vague since you have not provided your own level of education or within what framework you do research.
Since it is generally possible for anyone to publish a paper, you will not need to go through an academic. It may, however, be a very good step to take since essentially no-one can write a fault-free paper and if you do not have the experience or research eucation, you will find it almost necessary to stand a chance with a journal. Such a "review" will most certanly weed out any major problems with your paper. A problem here may of course be that your inexperience may make the task to improve the paper very time consuming (again, not knowing your own background makes it difficult to assess what help you may need) and hence make persons less likely to take on the job.
The journal will as I can see ot spend more or less time your manuscript than they would any other of similar quality. If the journal accepts it for review (general quality check) it will be sent out to reviewers for more careful scrutiny (scientific quality check) after which the paper is recommended for either revisions or is rejected. none of this would be different in your case unless one taes into consideration that you may not be as fluent in scientific writing (which means rejection is more liekly than otherwse).
So a good plan is to try to contact persons at an academic institution, present your work and see if they would consider helping you. It is not certain you will be helped because your manscript will be extra work done in addition to everything else. If you cannot get help then you can hopefully try to submit the manuscript anyway, but I strongly recommend you to spend time looking at how papers in the field are written and also pick up a good reference book on scientific writing to make sure you avoid basic mistakes.