Women who are in Christ have real and actual spiritual dominion over the beast/serpent - yes or no?
Yes
Why would women have fear or trepidation, but rather desire protection from him?
Sin (anything not done in faith is sin)
Does that make a woman broken to desire protection when she has been given dominion, or does only man have dominion over the serpent?
Not sure I understand the question but it is not "broken" to desire protection, even when already having been provided dominion by the cross. Eve was present and God spoke to both Adam and her when He said,
Genesis 1:28-30
God blessed them; and God said to them, “Be fruitful and multiply, and fill the earth, and subdue it; and rule over the fish of the sea and over the birds of the sky and over every living thing that moves on the earth.' Then God said, “Behold, I have given you every plant yielding seed that is on the surface of all the earth, and every tree which has fruit yielding seed; it shall be food for you; and to every beast of the earth and to every bird of the sky and to every thing that moves on the earth which has life, I have given every green plant for food”; and it was so.
Both were given the power and authority to accomplish the task set before them. This is called the "
dominion mandate," or "
cultural mandate," and, to my knowledge, it has never been revoked. It has, in fact been reworded and reiterated by God multiple time in the Bible's history. One problem is that Adam and Eve were good and sinless people and no one born after
Genesis 3:6-7 has been good or sinless. As a consequence a struggle exists within each of us in which the flesh contests the indwelling Spirit. For example, because the mind of flesh is hostile to God and cannot please God (Rom. 8:6)
using the mind of Christ we have already been provided (
1 Cor. 2:16), having the Spirit perform God's will in us (
Php. 2:13) is critically necessary. The ability to walk and work in the sovereignty of Christ is dependent upon the sanctifying work of the Spirit and our practice of what are commonly called the "
spiritual disciplines"
(Bible reading, prayer, fellowship, worship, teaching, service, evangelism, etc.). It is possible for a newborn Christian to walk in the nobility, authority, and power granted by Christ but not common. Practice makes perfect and maturity takes time and no one will be completely either on this side of the grave. I have always assumed the experience is part of the divine plan because God snap His fingers and make us perfect now but in His infinite wisdom chose not to do so from eternity.
Another problem is that Satan is also dead and enslaved by sin
but he has not option, no privilege for salvation available to him. In that regard we might consider what humanity would look like were it not for the redemptive work of God in creation, work so pervasive even the angels are assigned to minister to those in Christ Jesus. This is relevant because Satan has had all his glory stripped from him and he's been relegated to roam the earth as a carrion eater (
1 Pet. 5:8 -
Satan cannot literally devour Christians). Imagine Satan as a bull in the proverbial China shop, except the bull is the Tasmanian devil of the Looney Toons cartoons; he runs amuck causing destruction wherever he goes, limited only by the sovereignty of the Creator (
who has assigned the creature his place and time and purpose in creation). When the sin-saturated recklessness of Satan meets with the sinful flesh a party of depraved dross ensues. When the sin-saturated recklessness of Satan meets the royally priestly power and might of God living within us all the works of Satan are defeated (Eph. 6:10-19). A lesser version of that happens anytime sinful flesh meets sinful flesh and there is a lot of sinful flesh on the planet, a lot more sinful flesh than there is sin-enslaved devils. We're supposed to be part of the solution, not the problem, but the struggle is very real. One need only read through the epistolary to see how messy life in Christ among Christians can be.
Which is one of the reasons why scripture is so adamant about starting with oneself. No one is likely to rule over anything, especially others, if they cannot control themselves. This is especially demanding because we are supposed to start that rule with
love. Love, like salvation by grace through faith, is one of the great paradoxes of Christianity. Multiplying and subduing the earth and ruling over it is not (solely) about physically procreating and raising Christians as parental stewards of those God provides us. It is very much about preaching the gospel (which is why evangelism is a critical part of ones on spiritual development) and understanding the gospel is about the love of God expressed through His resurrected Son who is now King. Any preaching of the gospel that leaves out Jesus' kingship is an incomplete gospel. He is Savior of some, but Lord of all. If Christ is not Lord of our individual life then we will not subdue and rule anything.
There is a school of theology called Dominionism and it is somewhat diverse. It is also more often than not misguided in some way so if you find yourself reading a dominionist writer check what you read with scripture. For example, I happen to enjoy reading the Reconstructionists over at americanvisiondotorg, but they are wrong to think creating or achieving a Christian geopolitical state will precede or usher in the return of Christ. You may also have heard of the "Seven Mountains" dominionism commonly associated with the New Apostolic Reformation (NAR). Sadly, there is a lot of misinformation about the seven mountains perspective and the NAR is often very wayward in their thought and practice (the kingdom is not and will not be achieved by politics absent tangible Christian love). Both groups like to assume prominent Christian leaders of the 1960s-70s, like Francis Schaefer or Loren Cunningham as their progenitors but I'm fairly confident both men would be critical of these movements even though both would readily rest on the Dominion Mandate. Schaefer, for one, spent a great deal of his time preaching and practicing the preeminence of love (and wrote a pair of books on the subject).
A better way to look at the relevance of the dominion mandate is to believe God has provided everything you need for you to conquer all the circumstances of life he brings to you when you step foot outside your doorway every morning (
Dt. 31:8;
Pr. 3:3-4;
Pr. 3:5-6;
Mt. 17:20;
Php. 4:13;
1 Jn. 5:4 and a host of other verse found from beginning to end throughout the Bible).
Romans 12 is all about Genesis 1:28, especially everything from verse 9 on. Our job is to love, love sovereignly, and be overcomers. That is what Jesus did. The next time he comes it won't be so pleasant for others.
1 Peter 2:8-12 ESV
They stumble because they disobey the word, as they were destined to do. But you are a chosen race, a royal priesthood, a holy nation, a people for his own possession, that you may proclaim the excellencies of him who called you out of darkness into his marvelous light. Once you were not a people, but now you are God’s people; once you had not received mercy, but now you have received mercy. Beloved, I urge you as sojourners and exiles to abstain from the passions of the flesh, which wage war against your soul. Keep your conduct among the Gentiles honorable, so that when they speak against you as evildoers, they may see your good deeds and glorify God on the day of visitation.
Imagine how different the planet might look if we acted like we believe what Peter wrote. You,
@Hazelelponi, are an adopted daughter of The Most High God and one day you may be called to
judge angels. Yours is the nobility of being a creature created in God's own image who despite this nobility is called to walk humbly before God and man. Many are not chosen at all, but
you were chosen by God Himself.
. 
.
