Your question is an interesting one. Let's begin with issue of 1 Cor. 11:14. I might suggest that we re-word your question, as Paul is talking about men in general in 1 Cor. 14 and not ministers in particular. Whatever Paul is saying here, it applies to men and not ministers alone.
There is a variety of opinion on this verse. Some take the view that Paul is stating that the very order of nature ethically asserts that it is shameful for a man to have long hair. In this view, long hair for a man, in and of itself, would be sin. Hodge and Kistemaker appear to adopt this view. They add that Jewish men would likely not have had long hair, unless under the Nazarite vow. Other commentaries take the view that long hair, per se, is not the issue here, but what it potentially represents—gender confusion. Those in this camp note a variety of sources that indicate that long hair was acceptable among Jewish men and others during the biblical period. Calvin and Thiselton take this view. "Historical records bear, that in all countries in ancient times, that is, in the first ages, men wore long hair" (Calvin's commentary on 1 Corinthians, 361). I clearly am more convinced by the latter view. This in large part due to the word "shame" (amatio) used here in 1 Cor. 11:14 and also in Romans 1:26. The latter reference likewise has to do with gender confusion, and I believe is consistent with Paul's concern with the Corinthians. In both churches, homosexuality was an in issue. Paul rebukes this as sinful, and calls it shameful. This is unambiguous in Romans 1:26, which even clarifies how nature "teaches" this point; it does so by allowing them to "receive in themselves the due penalty for their error" (Romans 1:27). This seems to unambiguously refer to the natural diseases and maladies that come from homosexual behavior. It appears to me that in 1 Cor. 11:14 Paul is addressing the fact that Greek men were expressing their homosexual tendencies through the culturally recognizable sign of wearing long hair. Calvin affirms this as well (362 of same commentary). Likewise, women were also expressing their gender confusion by having their hair cut. I am personally not convinced that Paul is saying that it is a sin for a man to have long hair in and of itself, any more than he is saying that it is a sin for a woman to have short hair. It strikes me as odd that often people express concern about men having long hair, but rarely does the subject of women having short hair come up. Perhaps that is because it has become more culturally normal for women to have short hair than for men to have long hair. But Paul clearly addresses both in 1 Cor. 11:14-16. To summarize my view, it seems that the analogy of scripture would inform us that Paul's real concern is not the length of hair per se, but what is being communicated by it in a given context. If gender confusion, whether homosexuality or insubordination is being communicated, this is shameful, as even nature teaches.
I would add to the discussion a little more about Jewish men. There is not much (if any) OT biblical support to the idea of Jewish men having short hair as an ethical rule. The Nazarite vow required men to stop cutting their hair, but does not really address how long or short it was before that. But I would raise this point rhetorically: if nature teaches that it is a shame for men to have long hair according to the understanding above, does it cease "teaching" this when men take the Nazarite vow? The law and nature teach the same thing, thus it would be odd for the law to institute a practice that violates the law of nature.
To step beyond the biblical picture, it is provocative to notice the way in which paintings of our Lord relate to this issue. They certainly don't settle the issue, but they raise some interesting questions. You are probably familiar with the pictures of Jesus with long hair and robe, short hair and a suit and tie, and even African American Jesus. I'm not a fan of any of these pictures, but the one thing they prove is that we tend to project our culture backward as we imagine people from the biblical world. Your question is phrased very well in that you are asking it biblically. My only point here is that we should be careful not to impose our culture on the Bible (I am sure you agree). As an example, it is interesting to me that in the Puritan/colonial period, men not only wore long hair, some of them who did not have the hair even wore wigs! While long hair was not a rule among them (I would be cautious to even call it the norm), it was certainly a respectable option for godly men. I am sure they were aware of 1 Cor. 11:14, and likely appreciated Calvin's interpretation.
So what does long hair for a man communicate now? I think this is an important question. If it clearly is associated with homosexuality, than it would be unwise for a minister or any Christian man to wear it that way. But I would be a long way from believing that this is the case today. Homosexual men, to my knowledge, are not identified by wearing long hair. In fact, short hair is likely more dominant than long (the "metro-sexual" look?). Length of hair, at least in our day, does not seem to send a clear ethical signal one way or another. Thus, I would argue that a man is free to have long hair, and a woman is free to have short hair.
There is one last thing I would add. I think motives are something to consider. In other words, what might motivate a man to have long hair? If it is pride, vanity, etc., that would be sinful or at least unwise. But I find 1 Corinthians 7:4 helpful here. In this passage, the husband "has authority" over the wife's body, and the wife "has authority" over the husband's body. The context seems to be physical intimacy. It is my sense that husbands and wives ought to care for and even adorn their bodies for no one other than their spouse. Our bodies are not our own. They belong the Lord foremost, and have been given as gifts to our spouses who "have authority" over our bodies. The implication here is that if a wife prefers her husband to wear his hair short, he should. The opposite is also true. Servant-hood is the pattern, even when it comes to how we care for and adorn our bodies.
In conclusion, if a man has long hair, it may or may not be appropriate. Homosexuality is always inappropriate (a "shame"), but long hair on a man does not always send that signal. There may be other reasons that make it inappropriate (or even appropriate.), like motives, ministerial context, etc.
I hope this is helpful.
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