Before we do anything, we must come to the understanding that Paul's writings to the congregation at Corinth cannot be thoroughly understood via our modern mindsets, which tend to take Scripture out of context either historically, idiomatically, culturally, etc.
It is, however, imperative to understand that women's hair was a common object of lust in Antiquity; and therefore, Eastern Mediterranean women were expected to cover their hair.
Paul was attempting to address a clash of culture between upper and lower classes concerning fashion, as the upper class women were eager to show off their fashionable hairstyles - something that clashed with the lower class concern about sexual propriety....
So, let's begin by discovering exactly what the Bible says about women's hair:
1 Corinthians 11: 3. And I desire that you know that the head of every man is the Mashiyach, and the head of the woman is the man, and the head of the Mashiyach is Elohim. 4. Every man who prays or prophesies with his head covered, dishonors his head. 5. And every woman, who prays or prophesies with her head uncovered, dishonors her head; for she is on a level with her whose head is shaven. 6. For if a woman be not covered, let her also be shorn; but if it be shameful for a woman to be shorn or shaven, let her be covered....13. Judge for yourselves, among yourselves; is it becoming that a woman pray to Elohim with her head uncovered? 14. Does not nature teach you that in a man, if his hair grows long, it is a reproach to him? 15. But for a woman, if her hair is abundant, it is a glory to her; for her hair is given to her for a covering.
 It's a shame for a man to grow long hair, if his goal is to look like a woman. Torah requires that there be a clear distinction between male and female (Deuteronomy 22:5). Each person must be clearly recognizable according to their natural birth gender. Long hair on a man is NOT the sole issue; those taking Nazarite vows wore long hair some for life (Samson) and others for a season. Paul himself took a Nazarite vow. Rav Shaul is teaching against pagan sexuality practices, of those who do things to make themselves appear as the opposite gender. This an issue of gender distinction and understanding the Image of Elohim in both male and female.
 Or rather, as an additional covering, i.e., on top of proper clothing for worship and having her head covered. This does not mean a woman's hair is in place of a head covering.
According to the passage above:
- It is disgraceful for women to shave their hair.
- "Abundant" hair is a glory.
- Women must pray with their heads covered; their hair does not count as a covering.
While "abundant" hair is a glory, this does not mean women cannot cut or curl their hair to make it look its best. Not everyone has the good fortune to be born with thick and/or luxurious hair, and there is nothing wrong with cutting it into a nice style that suits our faces and figures. We are simply not to cut our hair into anything resembling a "masculine" style. Our bodies are, after all, YHWH's vessels and we are commanded to take care of them, from top to bottom and the inside-out (1 Cor. 6:19):
1 Corinthians 6: 19. Or don't you know, that your body is the temple of the Ruach haKodesh who abides in you, whom you have received from Elohim? And you are not your own. 20. For you are bought with a price. Therefore you glorify Elohim with your body, and with your spirit, which are Elohim's.
Just like the Bible does not say women have to wear dresses (after all men wore "dresses" in the old days before pants were invented!), the Bible also does NOT say they cannot cut their hair. It just says women dishonor themselves if they shave their heads, as we saw above.
There are no discussions in the Bible concerning the dyeing of hair because that wasn't an issue until modern times. However, there are many references that suggest "elders" are wise men and women to be respected; that the "changing seasons" of our lives are normal and natural, and that growing old is basically a badge of honor, including the graying of our hair:
Proverbs 16: 31 Gray hair is a crown of glory; it is gained in a righteous life.
Since the Bible does not prohibit changing the color of our hair, in this particular case, it is probably safe to say that it's up to each individual to decide what to do. YHWH wants us to take care of ourselves and look our best (we are, after all His vessels!), and if dyeing our hair makes us look better, there seems to be no biblical reason why we shouldn't - unless, of course, one is going for something unnatural, such as people who dye their hair purple and green, etc. in their attempt to "stand out" and draw attention to themselves. (We must remember, this life is not all about us; it's about God and our preparation for meeting Him one day - spiritual preparation that includes staying in the Word and learning to obey His Torah.)
YHWH has always been adamant about the roles and conduct of each gender. He made it very clear that women are to "act feminine" and dress modestly:
Deuteronomy 22: 5 "A woman is not to wear men's clothing, and a man is not to put on women's clothing, for whoever does these things is detestable to ADONAI your God.
1 Timothy 2: 9. So also, that women (appear) in a chaste fashion of dress; and that their adorning be with modesty and chastity; not with curls or with gold, or with pearls, or with splendid robes; 10. but with good works as is becoming to women who profess reverence for Elohim.
1 Peter 3: 3. And don't adorn yourselves with the external ornaments of curls of the hair, or of golden trinkets or of costly garments. 4. But adorn yourselves in the hidden person of the heart with a mild and pure spirit, an ornament that is precious before Elohim. 5. For so also the Set Apart women of old who trusted in Elohim, adorned themselves and were subject to their husbands....
The above obviously shows that the wearing today's micro-miniskirts and revealing cleavage, etc., is not allowed or acceptable!
An interesting side note borrowed from the Aramaic English New Testament:
Torah teaches that the Image of Elohim is both male and female (B'resheet/Gen. 1:27), and we learn about the unique and unified female and male attributes of YHWH in the Netzer of Isaiah 11:1-4. Rav Shaul also addressed the nature of YHWH as male and female in Romans 1:26, 27 when he spoke of male and female "natures" which are part of the Image of Elohim. The male/female characteristics are qualities that bind male and female partners together, but in the fallen world these unique differences are used to divide and destroy.
The uniqueness of male and female attributes is heavily guarded within Torah. There are many Commandments that establish clear gender recognition, which shows distinct honor to both genders while upholding the Creator's male and female characteristics. Clearly YHWH has chosen to reveal Himself as a Father with a predominately masculine identity, but He also reveals Himself through both male and female spirits. The marriage of male and female is a picture of the divine unity within YHWH and His people. Marriage between man and woman is Set Apart unto YHWH and to be guarded by each follower of Mashiyach. It is paganism to uphold either male or female attributes over the other because they are equal; but, it is also paganism to displace the Set Apart roles of male and female that YHWH established by His Word.