My answer is based solely from Tanakh as this question is about the intentions of God, rather than p'sak halakhah. I will also state that I interpret Tanakh using Jewish commentaries with an intention of finding internal consistency with Tanakh. So for example, if Rambam says a lot of stories in Tanakh are dreams, and this commentary requires a lot of reinterpreting of stories, a lot of further commentary on stories, and an explanation as to why the characters in stories don't perceive these events as dreams, then I would not accept Rambam's premise. My focus would be to understand the stories in the way the Tanakh itself, or the characters in Tanakh, relate to the stories.
So God provides dominion over the earth to humans in Genesis 1:28. The best way to understand God's intentions is to look for hints in the story itself and all closest stories to this event to see what we can glean. Here's what I've noticed.
1 - We never hear about Adam and Eve killing animals for any reason, even when they are naked they make clothes out of plants, it's God Himself who provides skin coverings. This makes me think that God did not intend for humans to kill animals for their needs, or at the very least, makes me think that Adam and Eve did not think God intended for them to kill animals for any reason.
2 - God parades animals before Adam for him to name them, and potentially choose a mate. I'm not going to pretend that I understand all of God's intentions here, but I see no evidence of killing animals for mankind's needs.
3 - Adam and Eve don't seem freaked out by the snake talking. Moshav Zekeinim brings the comment that humans and animals all spoke Hebrew until the tower of Babel. I find this commentary to be credible as it allows me to read all stories of animals talking without needing additional commentaries, explanations, interpretations, etc. If this commentary is true, mankind's subjugation could have been devoid of any trapping of animals. Instead, Adam and Eve would be able to talk to animals, and then ask them "to build and to uproot and plant and to quarry copper from its hills and [to do things] similar to this. And this is [all] included in its stating (above, verse 26), "and over all the earth."" It is my opinion that farm animals are definitely what God intended with dominion. I say this because God establishes many laws about taking care of farm animals, on how they should be treated while they're working, etc etc.
4 - Cain and Able are the next animal story as Able sacrifices animals. It's worth mentioning that Rabbi Albo thinks that Cain thought animal sacrifice was a sin, and therefore was appalled that God would accept the sacrifice of Able. This lends further evidence toward the idea that mankind did not see violence or the killing of animals as something God intended for mankind's dominion.
"Why did God not look favorably on Cain’s offering of his field
produce? The answer is that Cain had grown up watching his father Adam
toil and sweat to farm his land, surviving on vegetables alone as God
had forbade him and his family to eat meat. Cain himself became a
farmer, because he believed that there could be no distinction between
the human and the animal, except in the fact that the human must work
for his produce, whereas the animals simply took what was wild.
However, because both humans and animals both ate the same foodstuff,
Cain reasoned that they where essentially at the same spiritual level.
Thus Cain brought an offering from his field, and was appalled at the
idea of an animal sacrifice. Cain’s notion of equality between humans
and animals eventually led him to murder his brother, for he saw Abel
kill an animal for a sacrifice, and reasoned that if a human can kill
an animal, than a human can kill another human…Thus God allowed Noah
to eat animals to reintroduce the moral distinction between the animal
and the human."
Source: https://www.sefaria.org/sheets/115782.1?lang=en&with=all&lang2=en
5 - After the flood of Noah, mankind is finally given permission to eat animals, and it's the first time the discussion of killing animals as a concept is detailed in scripture. But despite permission being given, this permission comes with a new instinct for animals. Animals will now, for the first time EVER, be afraid of humans. Before this, trapping an animal would have been simple as walking up to it and tying a rope around it. If you're telling me God's intention was always for us to have to use force to subjugate animals, then this "instinct of fear" should have been given to animals in the beginning, but it wasn't.
Based on all these small details we see in the text regarding humans and animals, I do not see the intentions you outlined at the beginning of your question. I do not claim to speak for God, so I won't say what God truly intended. But I do find a lack of Biblical support for "Part of the point of Hashem giving the Earth into our domain is a permission for us to trap, subjugate and kill animals that would harm us or be pests for our crops and homes"
Based on everything I've outlined in scripture, I would choose to interpret " ורדו/subjugation" as employ. God made animals friendly to us, maybe even made them able to talk with us, and even in later years made rules about how to be good employers to animals (not yoking two different animals, not muzzling an animal who's working so they can eat while they work, etc etc). It seems to me that at the beginning, Adam and Eve interpreted God as wanting us to get along with, and employ animals for our purposes for raising food and resources. This subjugation required no force and no shedding of blood. If I could pick some modern categories, I think seeing eye dogs, rats that find land mines, and any animals that are trained peacefully for the well being of humans, are the closest examples of this subjugation/employment.
But the overall reality is this peaceful subjugation dynamic between humans and animals changes after the flood, and might further change after the tower of Babel. After those events, I do not think humans are capable of subjugating animals in the same way before. And at this point, the only way we can subjugate might be through force, violence, trapping, etc. But I would say this is different than what God intended, because again, it seems like God never even intended for animals to be scared of us for any reason.
Update
The OP asked the following to me in comments: "How do you understand ה֚וּא יְשׁוּפְךָ֣ רֹ֔אשׁ וְאַתָּ֖ה תְּשׁוּפֶ֥נּוּ עָקֵֽב from 3:15"
I understand this literally. God/Scripture punishes the snake for betraying Adam and Eve. One thing to keep in mind about these stories is that they take certain things for granted. So for example, the story takes for granted that animals "know" they are supposed to eat plants and vegetables, just as God commanded them. The story also takes for granted that the animals know that humans are supposed to have dominion over them, which is why they go to Adam to be named. The story also takes for granted that when God commanded Adam not to eat from the tree, this was not private knowledge. Eve who is theoretically created later knows about this rule, and so does the snake. Because the snake knows that humans were given dominion by God, and because the snake was named by Adam and was passed over as a mate, and because the snake knows about the one negative commandment given to Adam, his actions are indeed a true betrayal. And because of this, God forces an unnatural enmity between humans and snakes, that does not exist amongst any other animals with humans. So after the Garden of Eden, humans have been given an instinct to attack snakes, and snakes attack humans, as a sign of this deep betrayal from the garden. If I had to take my guess, the snake wanted God to take away dominion from Adam and Eve and give it to the serpent. Which is why God punishes the snake to become the lowest and most cursed of beasts, as a consequence of him wanting to raise himself to the highest of beasts, befitting of his great cunningness (Genesis 3:1)
One last comment I'll mention is that because animals are expected to know we have dominion, God makes it legal for us to kill animals that kill humans as stated in Genesis 6:
"וְאַ֨ךְ אֶת־דִּמְכֶ֤ם לְנַפְשֹֽׁתֵיכֶם֙ אֶדְרֹ֔שׁ מִיַּ֥ד כׇּל־חַיָּ֖ה אֶדְרְשֶׁ֑נּוּ וּמִיַּ֣ד הָֽאָדָ֗ם מִיַּד֙ אִ֣ישׁ אָחִ֔יו אֶדְרֹ֖שׁ אֶת־נֶ֥פֶשׁ הָֽאָדָֽם׃ But for your own life-blood I will require a reckoning: I will require it of every beast; of humankind, too, will I require a reckoning for human life, of everyone for each other!
שֹׁפֵךְ֙ דַּ֣ם הָֽאָדָ֔ם בָּֽאָדָ֖ם דָּמ֣וֹ יִשָּׁפֵ֑ךְ כִּ֚י בְּצֶ֣לֶם אֱלֹהִ֔ים עָשָׂ֖ה אֶת־הָאָדָֽם׃ Whoever sheds human blood,
By human [hands] shall that one’s blood be shed;
For in the image of God
Was humankind made. "
This text reads as if the animals are supposed to intrinsically understand that humans were made in the image of God and therefore they should have a tendency to avoid killing us. I imagine this viewpoint would be like great white sharks. We all know they can eat people, but the reality is sharks would prefer to eat anything other than humans first. We see that scripture views animals as having an understanding of the wrongness of killing humans because of the law of the goring ox.
Exodus 21:28-29
וְכִֽי־יִגַּ֨ח שׁ֥וֹר אֶת־אִ֛ישׁ א֥וֹ אֶת־אִשָּׁ֖ה וָמֵ֑ת סָק֨וֹל
יִסָּקֵ֜ל הַשּׁ֗וֹר וְלֹ֤א יֵאָכֵל֙ אֶת־בְּשָׂר֔וֹ וּבַ֥עַל הַשּׁ֖וֹר
נָקִֽי׃ When an ox gores a man or a woman to death, the ox shall be
stoned and its flesh shall not be eaten, but the owner of the ox is
not to be punished. וְאִ֡ם שׁוֹר֩ נַגָּ֨ח ה֜וּא מִתְּמֹ֣ל שִׁלְשֹׁ֗ם
וְהוּעַ֤ד בִּבְעָלָיו֙ וְלֹ֣א יִשְׁמְרֶ֔נּוּ וְהֵמִ֥ית אִ֖ישׁ א֣וֹ
אִשָּׁ֑ה הַשּׁוֹר֙ יִסָּקֵ֔ל וְגַם־בְּעָלָ֖יו יוּמָֽת׃ If, however,
that ox has been in the habit of goring, and its owner, though warned,
has failed to guard it, and it kills a man or a woman—the ox shall be
stoned and its owner, too, shall be put to death.
How do we know the animal is supposed to know better? One piece of evidence is that this similar law exists in other near eastern law codes, but the animal is not put to death, it is seemingly free of culpability.
Two differences that stand out are that in the Torah, the offending
animal is killed and if the animal is a habitual gorer, the owner is
killed as well. The Babylonian codes never kill the ox or the owner.
The difference is grounded in the gravity with which the Torah views
homicide: according to Numbers 35:33, killing a person pollutes the
land with bloodshed and so endangers those who reside there unless the
murderer is executed. This holds true even when the culpable party is
not human (Genesis 9:5).
Source: https://www.thetorah.com/article/the-law-of-the-goring-ox-is-it-neutered
The other piece of evidence is that the very next pasuk discusses the human owner with two situations: The first situation is that the human owner doesn't know the animal will gore other humans, and therefore is free from punishment. The second situation is that the human owner knows better, and if they know better, they receive the same punishment as the bull.
