Yes, I think you can ask for advice on your research statement from anyone who is A) qualified to give such advice (qualifications would include being someone who reads these statements to make hiring decisions, or at least someone who has successfully gotten a position that required them to write such a statement), and B) invested in your success and willing to gift you the time necessary. I would not send your research statement to people you do not know well and are not invested in you personally, unless it's part of an application for an open position.
Your advisor should certainly be covered in both of these categories, though it's possible few other people fit also into category (B).
People writing you rec letters might be good people to consider as well, but it doesn't seem like you've asked them for feedback. They have asked for your statement so that they can write you an appropriate recommendation letter; it would be destructive to your chances if they wrote about how you were an excellent candidate for a position in underwater basket weaving while you wrote a research statement about how you plan to do work in cheese making. If you want feedback, you'll need to ask for feedback.
The same goes for your advisor, too; if you have specific questions about your research statement, ask them about those specifics rather than just handing them the statement. Otherwise, it may not be clear whether you are looking for rigorous feedback or simple praise and encouragement.
Your question and comments reveal what seems to me to be a bit of a paranoid attitude towards these things. I understand that it can be very difficult to do an assignment when there is no "rubric" to know how you are graded, but hopefully by the end of your PhD you would have lost some of these notions that I think come primarily from earlier schooling where many assignments are graded on rigid criteria. In the "real world", even the "real world" aspects of academia, including graduate admissions as well as job applications for post doc and professor positions, there is no rubric. Your research statement is going to be read by the specific person/committee doing the job of hiring you. A good research statement is one that they like: specifically, one that is part of the overall total application that they like better than any other that they receive. There is no magic length, no special words, nothing cheap or easy that you can add to your statement to make it a guaranteed success. Different people and different committees will come to different conclusions when presented with exactly the same materials.